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ON SOME APPROACHES TOWARDS NON-COMMUTATIVE

ALGEBRAIC GEOMETRY

SNIGDHAYAN MAHANTA

Abstract. The works of R. Descartes, I. M. Gelfand and A. Grothendieckhave convinced us that commutative rings should be thought of as rings offunctions on some appropriate (commutative) spaces. If we try to push thisnotion forward we reach the realm of Non-commutative Geometry. The con-fluence of ideas comes here mainly from three seemingly disparate sources,namely, quantum physics, operator algebras (Connes-style) and algebraic ge-ometry. Following the title of the article, an effort has been made to providean overview of the third point of view. Since naive efforts to generalize com-mutative algebraic geometry fail, one goes to the root of the problem and triesto work things out “categorically”. This makes the approach a little bit ab-stract but not abstruse. However, an honest confession must be made at theoutset - this write-up is very far from being definitive; hopefully it will provideglimpses of some interesting developments at least.

February 1, 2008

1. Introduction

It behooves me to explore a little bit of history behind the subject . However, itwill be short as this is not meant to be a discourse on mathematical history and,in the process, significant contributions of several mathematicians spanning morethan two decades of mathematical research will have to be overlooked. Sincereapologies are offered to them.

In a seminal paper entitled Faisceaux algebriques coherents [Ser55] in 1955 Serrehad introduced the notion of coherent sheaves. Hindsight tells us that the seedsof non-commutative algebraic geometry were sown in this work, though it is notclear if Serre actually had such an application in his mind. In fact, it was Maninaround 1988 [Man88], who first propounded the idea of dropping the commutativityhypothesis. Pierre Gabriel in his celebrated work on abelian categories [Gab62] in1962 had also proved some reconstruction theorems for noetherian schemes, whichwere general enough to have an appeal to the mathematicians looking to go beyondthe commutative framework. Apparently, the real upsurge in such activities startedtaking place roughly 20 years back, with the works of Artin and Schelter on theclassification of non-commutative rings into some non-trivial classes. Of course, itwas not entirely an aimless pursuit as quantum physics had been a steady sourceof problems of non-commutative nature. There seems to be a plethora of proposalson the way to conduct research in this area. One of the most successful approaches

This is a “Diplom” thesis carried out under the patronage of Max-Planck-Institut fur Mathe-matik, Bonn. It is just an informal cruise through the subject and no originality, in any form, isclaimed.

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2 SNIGDHAYAN MAHANTA

was provided by Connes (see [Con94] and [Man91] for a broader perspective) butthat will be outside the purview of our discussion as we have an algebraic epithetin our title. Some more historical anecdotes and trends in current research can befound in the last chapter. Now we will take a brief look at the contents of eachchapter.

In chapter 1 we take a look at the description of Artin-Schelter regular algebrasof dimension 3 after Artin, Tate and Van den Bergh. We see how they arise fromtriples (E, σ,L ), which are well studied objects in commutative algebraic geometry.

In chapter 2 we almost entirely dedicate ourselves to Rosenberg’s reconstructionof varieties from their categories of quasi-coherent sheaves. In this, rather arid,excerpt we familiarize ourselves with Grothendieck Categories and find out how onecan associate a ringed space to such a category.

In chapter 3 an effort has been made to understand the preliminaries of themodel of non-commutative projective geometry spearheaded by Artin and Zhang.Towards the end we also include a discussion on non-commutative proper varietieswhich is a cumulative effort of works done by many different mathematicians.

In chapter 4 we provide, a rather sketchy, outline of a model of non-commutativealgebraic geometry initiated by Laudal. It has the advantage of having a goodtheoretical basis for working infinitesimally but it does not seem as natural as themodel described in chapter 3. Unfortunately we cannot even describe projectivegeometric objects in this approach as getting to the affine ones is quite an uphilltask in itself.

In chapter 5 we shall catch a glimpse of some other approaches in this directionwhich have had varied levels of success.

Finally, it must be bourne in mind that it is a brief survey of non-commutativealgebraic geometry and hence, as asserted earlier, very far from being exhaustive.References to the suitable articles have been provided amply and their ubiquity,coupled with the general expository nature of this write-up, account for the lack ofproofs. One final remark: this tome (except for chapter 1) is redolent of or stinksof categories as one’s tastes might be.

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In a paper entitled Some Algebras Associated to Automorphisms of EllipticCurves Artin, Tate and Van den Bergh [TVdB90] had given a nice descriptionof non-commutative algebras which should, in principle, be algebras of functions ofsome nonsingular “non-commutative schemes”. In the commutative case, nonsin-gularity is reflected in the regularity of the ring. However, this notion is insufficientfor non-commutative purposes. So Artin and Schelter had given a stronger regu-larity condition which we call Artin-Schelter(AS)-regularity condition. The mainresult of the above mentioned paper says that AS-regular algebras of dimension 3(global dimension) can be described neatly as some algebras associated to automor-phisms of projective schemes, mostly elliptic curves. And also such algebras areboth left and right Noetherian. This sub-section is entirely based on the contentsof [TVdB90].

To begin with, we fix an algebraically closed field k of characteristic 0. Weshall mostly be concerned with N-graded k-algebras A = ⊕

i>0Ai, that are finitely

generated in degree 1, with A0 finite dimensional as a k-vector space. Such algebrasare called finitely graded for short, though the term could be a bit misleading atthe first sight. A finitely graded ring is called connected graded if A0 = k. A+

stands for the two-sided augmentation ideal ⊕i>0

Ai.

Definition 1.1. (AS-regular algebra)A connected graded ring A is called Artin-Schelter (AS) regular of dimension d

if it satisfies the following conditions:(1) A has global dimension d.(2) GKdim(A) < ∞.(3) A is AS-Gorenstein.

It is worthwhile to say a few words about Gelfand-Kirillov dimension (GKdim)and the AS-Gorenstein condition of algebras.

Take any connected graded k-algebra A and choose a finite dimensional k-vectorspace V such that k[V ] = A. Now set FnA = k +

∑ni=1 V

i for n > 1. Thisdefines a filtration of A. Then the GKdim(A) is defined to be

GKdim(A) = limsupn

ln(dimkFnA)

ln(n).

Of course, one has to check that the definition does not depend on the choice ofV .

Remark 1.2. Bergman [KL85] has shown that GKdim can take any real numberα > 2. However, if GKdim 6 2, then it is either 0 or 1.

There are some equivalent formulations of the AS-Gorenstein condition availablein literature. We would just be content by saying the following:


Definition 1.3. (AS-Gorenstein condition)A connected graded k-algebra A of global dimension d <∞ is AS-Gorenstein if

ExtiA(k,A) = 0 for i 6= d and ExtdA(k,A) ≃ k

All regular commutative rings are AS-Gorenstein, which supports our convictionthat the AS-Gorenstein hypothesis is desirable for non-commutative analogues ofregular commutative rings. Further, note that the usual Gorenstein condition (forcommutative rings) requires that they be Noetherian of finite injective dimensionas modules of themselves.

Now we take up the task of describing the minimal projective resolution (⋆) ofan AS-regular algebra of dimension d = 3. As a fact, let us also mention that theglobal dimension of a graded algebra is equal to the projective dimension of the leftmodule Ak. Let

0 −→ P d −→ . . .f2−→ P 1 f1

−→ P 0 −→ Ak −→ 0(1)

be a minimal projective resolution of the left module Ak. P 0 turns out to beA; P 1 and P 2 need a look into the structure of A for their descriptions. SupposeA = T/I, where T = kx1, . . . , xn is a free associative algebra generatedby homogeneous elements xi with degrees l1j (also assume that x1, . . . , xn is aminimal set of generators). Then

P 1 ≈n⊕j=1

A(−l1j)(2)

The map P 1 −→ P 0, denoted x, is given by right multiplication with the columnvector (x1, . . . , xn)t.

Coming to P 2, let fj be a minimal set of homogeneous generators for the idealI such that deg fj = l2j . In T , write each fj as

fj =∑

j

mijxj(3)

where mij ∈ Tl2i−l1j . Let M be the image in A of the matrix (mij). Then

P 2 ≈ ⊕jA(−l2j)(4)

and the map P 2 −→ P 1, denoted M , is just right multiplication by the matrixM .

In general, it is not so easy to interpret all the terms of the resolution (1).However, for a regular algebra of dimension 3, the resolution looks like

0 −→ A(−s− 1)xt−→ A(−s)r

M−→ A(−1)r

x−→ A −→ Ak −→ 0(5)

where (r, s) = (3, 2) or (2, 3). Thus such an algebra has r generators and rrelations each of degree s, r + s = 5. Set g = (xt)M ; then

gt = ((xt)M)t = QMx = Qf(6)


for some Q ∈ GLr(k).Now, with some foresight, we introduce a new definition, that of a standard

algebra, in which we extract all the essential properties of AS-regular algebras ofdimension 3.

Definition 1.4. An algebra A is called standard if it can be presented by r genera-tors xj of degree 1 and r relations fi of degree s, such that, with M defined by (3),(r, s) = (2, 3) or (3, 2) as above, and there is an element Q ∈ GLr(k) such that(6) holds.

Remark 1.5. For a standard algebra A, (5) is just a complex and if it is a resolution,then A is a regular algebra of dimension 3.

Multilinearization and a Moduli Problem

We have some algebraic objects, namely non-commutative k-algebras, and wewant to associate some geometric data to them. Ideally we would like to think of thealgebra as the “coordinate ring” of some “space”, but this is a utopia at the moment.Nevertheless, we can associate some other geometric objects to them, namely afamily of schemes Γd, which is called the multilinearization of the algebra. Nowwe shall try to understand the interplay between the two.

N− graded non− commutative k − algebras←→ families of schemes Γd

However, the figure above is not as nice as it looks; the arrow is not reallyreversible. If one starts with an algebra A and passes on to its multilinearizationand then tries to recover the algebra from it, one gets another algebra B togetherwith an algebra map A −→ B which is bijective in degree 1.

Algebra to Family of Schemes

Let A = T/I be a graded k-algebra, where T = k(x0, . . . , xn) is the free asso-ciative algebra generated by x0, . . . , xn of degree 1 and I is a homogeneous ideal ofT .

Let V = T1 and set P = P(V ) = Pnk . LetO(1, . . . , 1) = pr∗1OP (1)⊗· · ·⊗pr∗dOP (1)denote the twisting sheaf on the product (fibre product over Spec k) of d copies of

P . Every homogeneous element f ∈ Td = T⊗d1 defines a global section f of thissheaf. Note that A was defined as a quotient of T by a homogeneous ideal I. LetId = f : f ∈ Id. Then we define

Γd = Z(Id) ⊂ Pd(7)

where Z denotes the scheme of zeros. Of course, with natural conventions we have,

˜(fg)(v1, . . . , vp, w1, . . . , wq) = f(v1, . . . , vp)g(w1, . . . , wq)(8)

for f ∈ Tp, g ∈ Tq

It must be mentioned here that actually the family Id is called the multilineariza-tion of A as it consists of multilinear forms on V but by abuse of notation, whichis very much in vogue, we call their scheme of zeros multilinearization as well. Let


us rush through some of the properties of this family of schemes so defined. Ratherformal arguments will lead to a proof of most of them [TVdB90] and hence will notbe provided here.

Proposition 1.6. 1. For any d, Γd+1 ⊂ (P ×Γd)∩ (Γd ×P ) and equality holds ifId+1 = T1Id + IdT1 (for instance if I is generated in degrees 6 d).

2. Let Pi denote the i − th factor of the product P d and for 1 6 i < j 6 d, let

prij = pr(d)ij denote its projection to the product

j∏ν=i

Pν . Then prij(Γd) is a closed

subset of Γj−i+1.

Proposition 1.7. Let 0 6 i 6 d and let πi : Γd+1 −→ Γd denote the projection(dropping the (i+ 1)− th component)

(p1, . . . , pi, pi+1, pi+2, . . . , pd+1) 7−→ (p1, . . . , pi, pi+2, . . . , pd+1)(9)

1. The fibres of πi are linear subspaces of P .2. Let p ∈ Γd, and let L be the fibre of πi at p. If dimL 6 0 (this encompasses

the fact that the fibre at a point could be empty), then πi is a closed immersionlocally in a neighbourhoud of p ∈ Γd.

Proposition 1.8. 1. Assume that for some d, pr(d+1)1d defines a closed immersion

from Γd+1 to Γd, thus identifying Γd+1 with a closed subscheme E ⊂ Γd. ThenΓd+1 defines a map σ : E −→ Γd, by the rule

σ(p1, . . . , pd) = (p2, . . . , pd+1)(10)

where (p1, . . . , pd+1) is the unique point of Γd+1 lying above (p1, . . . , pd) ∈ Γd.2. If, in addition to this, σ(E) ⊂ E and if I is generated in degree 6 d, then

pr(n)ij : Γn −→ E is an isomorphism for every n > d+ 1.

Now we introduce the definition of point modules.

Definition 1.9. A graded right A-module M will be called a point module if itsatisfies the following conditions:

1. M is generated in degree zero.2. M0 = k, and3. dimkMi = 1 for all i > 0.By adding one more adjective, namely truncated, we arrive at the notion of a

truncated point module. A truncated point module of length d+ 1 is a point modulewhose Hilbert function is truncated, i.e.,

dimMi =

1 if 0 6 i 6 d,

0 otherwise.

Put differently, a point module can be thought of as a module, generated in degreezero, whose Hilbert series is (1 − t)−1 (and for a truncated point module it is justthe polynomial 1 + t+ · · ·+ td).


Our next aim is to unravel the following statement:

Γd represents the functor of flat families of truncated point modules oflength d+ 1 parametrized by SpecR.

As a digression let me say a few words about moduli problems. For reasons notvery hard to see, it is desirable to come up with a “space” which parametrizescertain families of objects defined over schemes modulo some equivalence relation.Let k be a field and let Schk denote the category of schemes over k (or Spec k ifyou like). Let F be a functor from Schok to Sets (category of sets),

F : Schok −→ Sets

B 7−→ F (B)(11)

where F (B) is the set of equivalence classes of families of objects parametrizedby the scheme B. Such an F is called a moduli problem. A typical example will bethe problem where we associate to a scheme the family of elliptic curves with ann-level structure modulo isomorphism or the Picard functor for a curve.

An ideal solution to a moduli problem would be a natural isomorphism Φ to thefunctor of points of a scheme, say M . If such an M exists, then it is called thefine moduli space of the corresponding moduli problem. Let me remind you thata scheme X is no more than its functor of points hX = Hom(−, X) by Yoneda’sLemma. But we are not always fortunate enough to be able to construct a finemoduli space for a moduli problem and so we define what is called a coarse modulispace. However, I am not going to go into these discussions over here.

It should also be bourne in mind that if X is a k-scheme, then Hom(SpecR,X),denoted X(R), is called the set of R-valued points of the scheme X , where R is acommutative k-algebra.

Having gone through the general set-up the first thing that we need to do is toformulate a notion of a family of point modules. We shall work over the categoryof affine k-schemes, which is equivalent to the category of commutative k-algebras,denoted Algk.

We once again turn back to our graded k-algebra A (not necessarily commu-tative). Let R ∈ Algk and consider a graded R ⊗k A-module M generated indegree 0 and satisfying M0 = R. By restriction of scalars via the canonical mapR −→ R ⊗k A, we consider M as an R-module. Since we are trying to define afamily of point modules, it is expected that locally i.e., for all p ∈ SpecR, Mp

behave like a point module. Hence, we also ask that for every degree d, Md (thesheaf associated to the graded piece Md of degree d) be a locally free sheaf of rank1 over Spec R.

To define a family of truncated point modules of length d + 1 we just ask thatMj be locally free of rank 1 for all 0 6 j 6 d and otherwise the stalk be 0 in somesense.

To phrase it in the moduli problem terms we say that our functor F associates toany affine scheme over k, Spec R, the isomorphism classes of all such R⊗kA-modulesM (the truncated version). One has to take the obvious notion of isomorphismamongst such modules.


At the end of all these discussions we may conclude that the statement in boldletters is equivalent to saying,

Proposition 1.10. Γd is the fine moduli space for the moduli problem F .

Sketch of the Proof: Our aim is to show that the R-valued points of Γd cor-respond to the isomorphism classes of flat families of truncated point modules oflength d+ 1, parametrised by SpecR.

We start with a family of truncated point modules of length d+ 1 parametrisedby SpecR. So, by definition this is a graded R⊗A-module M , generated in degreezero, such that M0 = R and that Mi is locally free of rank 1 for each i, 1 6 i 6 d.Now choose a covering SpecR = ∪

jSpecRfj , such that each (Mfj )i is a free Rfj -

module of rank 1. To simplify notations, let me assert right here that if we canfind SpecRfj −→ Γd for each j, then they will glue properly to give rise to a mapSpecR −→ Γd and hence, an R-valued point of Γd. So we might as well pretendthat R were equal to one of its localizations and hence, “drop the subscripts fj”.

We have a collection of free R-modules of rank 1; we promptly choose a basisfor each one of them. Let mi be the chosen basis element for Mi. If you have allour hypotheses at your fingertips, you can easily recall that A was generated byxj , 0 6 j 6 n, which were elements of degree 1. Now we write out the products ofthe bases mi by xj .

mi−1xj = miaij(12)

for some aij ∈ R. In this way we obtain a set of d points

ai = (ai0, . . . , ain) ∈ P ∀ 0 6 i 6 d(13)

Evidently a = (a1, . . . , ad) gives an R-valued point of P d. But I want to claimthat actually a ∈ Γd(R). Take any a ∈ P d(R) = Hom(SpecR, P d) ≃ Hom(Td, R)

obtained in the manner described above. Let me remind you that Td = T⊗d1

corresponds to the scheme P d. Now for any element f ∈ R⊗T of degree d, simplyby looking at the definition of ai we can convince ourselves that

m0f = mdf(a)(14)

But if f ∈ R⊗ Id then f = 0 in R⊗A and hence f(a) = 0. And Γd was defined

precisely as the scheme of zeros of Id. So a is indeed an R-valued point of Γd.

[If the symbols confuse you, let me put it for you differently. The element a shouldbe a map from T −→ R. Take any f ∈ Id ⊂ Td. It is nothing but a polynomialin x0, . . . , xn with coefficients in k, which we denote f . Tensor it with R to obtaina polynomial with coefficients in R. Now “evaluate” it at a. This evaluation maphas a kernel and that is precisely Id (or Id if you like).]

Now the passage from Γd to M or schemes to modules is just a matter of walkingback. We start with a point of Γd(R) i.e., ρ : SpecR −→ Γd. Then consider thefollowing sheaf.


Lj = i∗O(1, . . . , 1︸︷︷︸j times

, 0, . . . , 0)

where i : Γd → P d, and put L =d⊕j=0

Lj .

We just pull back this sheaf L on Γd via ρ to obtain the desired truncatedpoint module. Imitating the authors, I allow myself the luxury of cutting somecorners and summing it up by asserting that the two “functors” defined above are“quasi-inverses” of each other.

Remark 1.11. This functor of points description of the family of schemes Γd saysthat it is intrinsic to the algebra A, and does not depend on the presentation of Aas a quotient of T .

The family Γd along with the maps π defined in [1.7] forms an inverse sys-tem and the inverse limit gives us an object whose “points” correspond to “pointmodules”.

From Family of Schemes back to Algebra

The most naive attempt to recover an algebra A from its multilinearisation Γdyields an algebra B and a canonical morphism A −→ B. In fact, we can associatecanonically a graded algebra B to any sequence of subschemes Zd ⊂ P d havingthe property

pr1,d−1(Zd) ⊂ Zd−1 and pr2,d(Zd) ⊂ Zd−1 ∀ d(15)

Let Ld = O(1, . . . , 1)⊗OPdOZd . Then set Bd = H0(Zd, Ld). Since pr1,i(Zi+j) ⊂

Zi and pri+1,i+j(Zi+j) ⊂ Zj , we have Zi×Zj ← Zi+j . Applying theH0 functor willjust reverse the arrows and define for us a multiplication mapBi×Bj −→ Bi+j usingthe isomorphism pr∗1,i(Li)⊗OPd pr

∗i+1,i+j(Lj) −→ Li+j . This makes B = ⊕

dBd into

a graded associative algebra. It must be mentioned that B need not be generatedin degree 1. If it is so, then the sequence Zd can be “properly contained” in themultilinearization of B, Γd(B).

Proposition 1.12. Let A = T/I be a quotient of a free associative ring and letΓd be the multilinearisation of A. Let B be the algebra associated to it. Then,there is a canonical morphism, φ : A −→ B, which is bijective in degree 1.

Proof:The functorial maps

H0(P d,O(1, . . . , 1)) −→ H0(Γd, Ld)(16)

define a morphism at the level of the graded pieces and hence a morphism fromthe free algebra T to B. This is just the restriction of sections. Once again thedefinition of Γd as the scheme of zeros of sections Id forces I to be in the kernel ofthis morphism and hence this map factors through A = T/I.


In degree 1, Γ1 is the dual projective space of A1 = T1/I1 and hence its sheaf ofglobal sections H0(Γ1, L1) = B1 ≃ A1 [Lefschetz Hyperplane Theorem]. (It is dualbecause V = T ∗1 and Γ1 = P((T1/I1)

∗) ⊂ P(V ) = P ).

This process of passing onto the multilinearization is rather unwieldy. Fortu-nately, we don’t need to consider the entire family of schemes. We can concentrateon a smaller piece of datum in the form of a triple (E, σ,L ), which will be taken upnow. The recipe to obtain the triple from an algebra (if at all possible) is explainedin the “main result” subsection.

Twisted Homogeneous Coordinate Rings

Here we provide a very general recipe of manufacturing interesting non-commutativerings out of a completely “commutative geometric” piece of datum, called an ab-stract triple, which turns out to be an isomorphism invariant for AS-regular alge-bras.

Definition 1.13. An abstract triple T = (X,σ,L ) is an assortment of a projectivescheme X, an automorphism σ of X and an invertible sheaf L on X.

It is time to construct the Twisted Homogeneous Coordinate Ring B(T ) out ofan abstract triple. For each integer n > 1 set

Ln = L ⊗Lσ ⊗ · · · ⊗L

σn−1

(17)

where L σ := σ∗L . The tensor products are taken over OX and we setL0 = OX . As a graded vector space, B(T ) is defined as

B(T ) = ⊕n>0

H0(X,Ln)

For every pair of integers m,n > 0, there is a canonical isomorphism

Lm ⊗k Lσm

n −→ Lm+n

and hence defines a multiplication on B(T )

H0(X,Lm)⊗kH0(X,Ln) −→ H0(X,Lm+n).

Example 1. Let us compute (more precisely, allude to the computation of) thetwisted homogeneous coordinate ring in a very simple case. Let T = (P1,O(1), σ),where σ(a0, a1) = (qa0, a1) for some q ∈ k∗.

We choose a parameter u for P1, so that the standard affine open cover U =P1 \∞, V = P1 \ 0 has rings of regular functions O(U) = k[u] and O(V ) = k[u−1].Now we can identify O(1) with the sheaf of functions on P1 which have at most asimple pole at infinity; in other words, it is a sub-sheaf of k(u) = k(P1) generatedby 1, u. It is now easy to see that H0(X,O(n)) is spanned by 1, u, . . . , un andthat, as a graded vector space B(P1, id,O(1)) = kx, y (the free algebra over kgenerated by x and y and not the usual polynomial ring), where x = 1 and y = u,thought of as elements of B1 = H0(X,O(1)). It should be mentioned that σ acts


on the rational functions on the right as fσ(p) = f(σ(p)) for any f ∈ k(P1) andp ∈ P1. It is clear from our concrete presentation that O(1)σ ∼= O(1). So as agraded vector space B(T ) ∼= B(P1, id,O(1). However, the multiplication is twisted.

y.x = y ⊗ xσ = u⊗ 1σ = u⊗ 1 = u ∈ H0(P1,O(2)).

On the other hand,

x.y = x⊗ yσ = 1⊗ uσ = 1⊗ qu = qu ∈ H0(P1,O(2)).

So we find a relation between x and y, namely, x.y − qy.x = 0 and a littlebit of more work shows that this is the only relation. So the twisted homogeneouscoordinate ring associated to T is

B(T ) = kx, y/(x.y − qy.x).

A good reference for a better understanding of these rings is [AVdB90].

The Main Result

The main result of the paper [TVdB90] asserts that the AS-regular algebras ofdimension 3 are precisely the non-degenerate standard algebras. Let me now devotesome time to the italicized words, which we have not seen yet and hopefully thereader will be convinced that they are more tractable objects and give us a betterunderstanding of the AS-regular algebras of dimension 3.

Let A denote a k-algebra which is presented in the form A = T/I, where T is anon-commutative polynomial ring with r generators of degree 1 and I is an idealgenerated by r linearly independent relations of degree s. We assume, moreover,that (r, s) = (2, 3) or (3, 2) keeping the definition of a standard algebra in mind,which in turn is invoked to concentrated on AS-regular algebras of dimension 3.

Let f = (f1, . . . , fr)t be the column vector of defining relations i.e., of generators

of I. There are uniquely defined matrices M,N with entries in the tensor algebraT such that

f = Mx and f t = xtN.(18)

Let P = Pr−1. By P d we mean P ×k · · · ×k P︸︷︷︸d times

. We consider the multilineariza-

tion Γd ⊂ P d. It follows from [1.8] that the whole sequence of schemes is de-termined by Γ = Γs. Set π1 = pr1,s−1 |

Γand π2 = pr2,s |

Γ(refer to [1.7]).

Now let E = π1(Γ) and E′ = π2(Γ). By definition, E is the locus of points(p1, . . . , ps−1) ∈ P s−1 for which there exists ps ∈ P such that (p1, . . . , ps−1, ps) ∈ Γ.

So E is the locus of zeros in P s−1 of the multihomogeneous polynomial detM = 0.Similarly, E′ is the locus of detN = 0. Thus E is either all of P s−1 if detM ≡ 0, oris a Cartier divisor in P s−1, and similarly for E′.

Definition 1.14. We call an algebra A semi-standard if the schemes E and E′ areequal in P s−1 or equivalently if

detM = c.detN(19)


for some c ∈ k∗.

The constant c is independent of the change of basis in x and f , provided thatdetM is not identically 0. As one can see, this definition does not really need theassumption that (r, s) = (2, 3) or (3, 2).

Now, let A be a semi-standard algebra, so that E = E′. We may view Γ as thegraph of a correspondence σ on E via the closed immersion (π1, π2) : Γ −→ E×E.It should be noted that π2 = π1 σ.

Definition 1.15. We call a semi-standard algebra non-degenerate if the sigmamentioned above is actually an automorphism of E, and degenerate otherwise.

If there is any sanity in the nomenclature, then one should expect standardalgebras to be semi-standard. This is indeed the case and its verification is just amatter of playing with the definitions.

The good news is that we are already in sight of an abstract triple. If A isa non-degenerate semi-standard algebra then we can associate the triple T (A) =

(E, σ,L ), where E = π1(Γ)i→ P s−1, σ is the automorphism given by the non-

degeneracy of the algebra and L is contrived from the twisting sheaf as ψ∗OP (1),where ψ : E −→ P = Pr−1 is the morphism defined by

ψ =

i : E −→ P2 if (r, s) = (3, 2),

pr1 i : E −→ P1 if (r, s) = (2, 3).(20)

It must be bourne in mind that (r, s) = (2, 3) or (3, 2) and hence, P s−1 =P1 ×k P1 or P2 respectively. Hence, ψ is determined by the sections of L . Thisprocedure of obtaining abstract triples can also be applied to any family of schemesΓd ⊂ P d, satisfying Γd ∼= Γs ∀ d > s and π1(Γs) = π2(Γs) ⊂ P s−1.

It follows from the definition that AS-regular algebras of dimension 3 are stan-dard. And it has also been shown in [TVdB90] that AS-regular algebras are non-degenerate. So we can associate a triple to such an algebra. The regularity of thealgebra is reflected in the following property [21] of the triple, which is also calledthe regularity property of the triple for obvious reasons.

(σ − 1)2[L ] = 0 if r = 3

(σ − 1)(σ2 − 1)[L ] = 0 if r = 2.(21)

where [L ] denotes the class of L in PicE. Regular algebras give rise to regulartriples, which should come as no surprise. But, in the paper it is also shown thatstandard algebras give rise to regular triples [1.16] and that algebras associated toregular triples are also AS-regular.

Let A′ be a non-degenerate semi-standard algebra and let T (A′) = (E, σ,L )be the triple associated to it, with E → P s−1. Let B = B(T ) denote the twistedhomogeneous coordinate ring constructed out of the triple T (A′). Then we defineanother algebra A = A(T (A′)). Set T = ⊕

n>0Tn be the tensor algebra over k on B1.

The definition of a tensor algebra gives us a canonical homomorphism T −→ B. LetJ = ⊕

n>0Jn be its kernel, and let I denote the two-sided ideal of T generated by Js.

We then put A(T ) = T/I. Then the composition of the natural homomorphisms


A(T ) = T/I −→ T/J −→ B = B(T )

gives us a canonical homomorphism A(T ) −→ B(T ), which is bijective in degree1 (since T1 = B1 by definition).

Theorem 1.16. The algebras A and A′ are canonically isomorphic. Further, if A′

is standard to begin with, then T (A′) is regular as a triple.

This theorem tells us that the AS-regular algebras of dimension 3 are indeed thesame as non-degenerate standard algebras. But non-degenerate standard algebrascome hand in hand with an abstract triple and hence are more amenable to algebraicgeometric tools. The story does not quite end here. Such algebras have one moredesirable property.

Theorem 1.17. Let T be an abstract triple. Then B = B(T ) is both left and rightnoetherian.

Hence, AS-regular algebras of dimension 3 are also both left and right noetherian.The proofs of these theorems are rather long and intricate. Hence, they could notbe incorporated, though some ideas could be instructive.


2. Towards Non-commutative Schemes via Grothendieck Categories

Whether we like it or not, one has to always do some mundane things at thebeginning i.e., fixing notations and conventions.

All categories, unless otherwise stated, will be assumed to be locally small. Notall arguments need this assumption but it just saves us some headaches. Also, sincewe are talking about non-commutative algebraic geometry, it is needless to say thatrings and algebras are not assumed to be commutative. It should also be addedthat unadorned terms like schemes and sheaves refer to the usual (commutative)ones.

Qcoh(X) := category of quasi-coherent sheaves on a scheme X .

Coh(X) := category of coherent sheaves on X .

Mod(A) := category of right A-modules, where A is a k-algebra.

Now we delve into some “philosophy”. In mathematics we generally try to keepthings tidy and organized even if it comes for a price, say at the level of complexity.To incorporate multifarious data we define increasingly complicated mathematicalobjects, their morphisms and their compositions and categories provide the frame-work where such things are practicable. Jumping into conclusion - categories areindispensable! One might look at the set of integers with the two operations (ad-dition and multiplication) just as a set or make an intellingent definition out ofthe whole lot of mess and call it a ring. Now to study a ring one might try toinvestigate all the questions in a myopic way looking only at the ring or map it tosimple modules and try to extract information from that. In the case of a PID thestructure theorem of the finitely generated modules over it already gives us a goodidea about the underlying structure of the ring. And now, armed with gut feeling,we take a leap in reasoning and say that the study of a ring can be “reduced” to thestudy of the category of modules over that ring. It is not entirely speculative as onecan get a part of the ring (its centre) back by taking the endomorphism ring of theidentity functor of the module category. As stated earlier, the category of rings isjust the opposite category of affine schemes and the category of modules is just thecategory of quasi-coherent sheaves. Mathematical history reveals that wheneverwe have many equivalent formulations of the same concept, we should not ignoreany one as there might be occasions where one notion generalizes, while the othersdo not. So if we cannot make sense of a scheme in the direct way (it is a truismthat in the non-commutative world we do not have “enough” two-sided ideals andhence localizations have very little chance of being pulled off), we pass on to thelevel of categories. If one is not too pedantic one should not complain if the follow-ing assertion is made - a non-commutative scheme should be an abelian categorywhich resembles the category of quasi-coherent sheaves on a commutative scheme.What are such categories?? Definitely they are not unwieldy arbitrary categories;we expect them to be at least abelian. Actually what we want are GrothendieckCategories invoking the paper by Grothendieck [Gro57], where he had written downthe famous AB Properties extracting geometric content out of arbitrary abeliancategories. At the moment this is like putting two and two together and gettingfive, but hopefully this choice will be vindicated by the time we reach the end of this


section. Cutting the long story short, we directly present the definition of, what wecall today, a Grothendieck Category, which will be our main object of interest forsome time at least.

Grothendieck Categories

Definition 2.1. Grothendieck CategoryIt is a locally small cocomplete (i.e., closed under all coproducts) abelian category

with a generator and satisfying, for every family of short exact sequences indexedby a filtered category I [i.e., I is non-empty and, if i, j ∈ Ob(I) then ∃ k ∈ Ob(I)

and arrows i −→ k and j −→ k, and for any two arrows iu−→−→v

j ∃ k ∈ Ob(I) and

an arrow w : j −→ k such that wu = wv (think of a categorical formulation of adirected set)].

0 −→ Ai −→ Bi −→ Ci −→ 0

the following short sequence is also exact

0 −→ colim−→i∈I

Ai −→ colim−→i∈I

Bi −→ colim−→i∈I

Ci −→ 0

i.e., passing on to filtered colimits preserves exactness. This is equivalent to thesup condition of the famous AB5 Property.

Remark 2.2. The original AB5 Property requires the so-called sup condition,besides cocompleteness. An abelian category satisfies sup if

for any ascending chain Ω of sub-objects of an object M , the supremum of Ωexists; and for any sub-object N of M , the canonical morphism

supL ∩N |L ∈ Ω∼−→ (supΩ) ∩N

is an isomorphism. Hence, another definition of a Grothendieck Category couldbe a cocomplete abelian category, having a generator and satisfying the sup condi-tion i.e., an AB5 category with a generator.

For the convenience of the reader let me say a few things about a generator ofa category. An object G of a category C is called a generator if, given a pair ofmorphisms f, g : A −→ B in C with f 6= g, there exists an h : G −→ A withfh 6= gh (more crisply, Hom(G,−) : C −→ Set is a faithful functor). A family ofobjects Gii∈I is called a generating set if, given a pair of morphisms f, g : A−→−→Bwith f 6= g, there exists an hi : Gi −→ A for some i ∈ I with fhi 6= ghi. Strictlyspeaking, this is a misnomer. In a cocomplete category, a family of objects Gii∈Iforms a generating set if and only if the coproduct of the family forms a generator.

Remark 2.3. Let C be a cocomplete abelian category. Then an object G is agenerator if and only if, for any object A ∈ Ob(C) there exists an epimorphism,

G⊕I −→ A

for some indexing set I.


Example 2. (Grothendieck Categories)

1. Mod(R), where R is an associative ring with unity.2. The category of sheaves of R-modules on an arbitrary topological space.3. In the same vein, the category of abelian pre-sheaves on a site T . Actually

this is just Funct(T op, Ab).4. Qcoh(X), where X is a quasi-compact and quasi-separated scheme. (a mor-

phism of schemes f : X −→ Y is called quasi-compact if, for any open quasi-compact U ⊆ Y , f−1(U) is quasi-compact in X and it becomes quasi-separatedif the canonical morphism δf : X −→ X ×Y X is quasi-compact. A scheme Xis called quasi-compact (resp. quasi-separated) if the canonical unique morphismX −→ Spec(Z), Spec(Z) being the final object, is quasi-compact (resp. quasi-separated)).

Grothendieck categories have some remarkable properties which make themamenable to homological arguments.

1. Grothendieck categories are complete i.e., they are closed under products.2. In a Grothendieck category every object has an injective envelope, in partic-

ular there are enough injectives.

The deepest result about Grothendieck categories is given by the following the-orem.

Theorem 2.4. [Gabriel Popescu] Let C be a Grothendieck category and let G be agenerator of C. Put S = End(G). Then the functor

Hom(G,−) : C −→Mod(Sop)

is fully faithful (and has an exact left adjoint).

We wrap things up in this episode by showing that Grothendieck categories are,in some sense, “big” amongst abelian categories. Let C be a small abelian category.Then define the category IndC as:

Ob(IndC) =Functors from all filtered categories to C.

For convenience we denote them Tjj∈J , where J is a filtered category.

HomIndC(Tii∈I , T ′jj∈J) = Limj∈J colimi∈I HomC(Ti, T′j)

By means of a dual construction one arrives at, what is called, a pro-categoryProC. The category IndC has a more tangible realisation in the form of a naturallyequivalent category, denoted Lex(C), whereOb(Lex(C)) = left exact additive functors Cop −→ Ab. By Yoneda’s theorem thefunctor

C −→ Lex(C)

A 7−→ HomC(−, A)

is fully faithful and exact. This gives an embedding of the abelian category C intoLex(C). It is known that if C is a small abelian category and D is a Grothendieckcategory, then Funct(Cop,D) is also a Grothendieck category. Hence, in particular,Lex(C) is a Grothendieck category. So we obtain an embedding of a small abelian


category into a Grothendieck category. Now we mention a theorem which will comein handy later on.

Theorem 2.5. [P. Gabriel] Let C be a noetherian abelian category. Then the cat-egories C and IndC determine each other up to a natural equivalence. In fact, C isthe full sub-category formed by all the noetherian objects in IndC.

If X is a noetherian scheme, then Ind(Coh(X)) ≃ Qcoh(X). Let Noeth denotethe operation of taking noetherian objects. Then, Noeth(Qcoh(X)) ≃ Coh(X). Sofor a noetherian scheme X , we might as well deal with the more tractable categoryCoh(X).

Justification for bringing in Grothendieck Categories

We begin by directly quoting Manin [Man88] - “...Grothendieck taught us, todo geometry you really don’t need a space, all you need is a category of sheaveson this would-be space.” This idea gets a boost from the following reconstructiontheorem.

Theorem 2.6. [Gabriel Rosenberg [Ros98]] Any scheme can be reconstructed uniquelyup to isomorphism from the category of quasi-coherent sheaves on it.

An easy reconstruction:Let us look at a very simple case where this theorem is applicable. When it

is known in advance that the scheme to be reconstructed is an affine one, we canjust take the centre of the category, which is the endomorphism ring of the identityfunctor of the category. More precisely, let X = SpecA be an affine scheme and letA be the category of quasi-coherent sheaves on X , which is the same as Mod(A).Then the centre of A, denoted End(IdA), is canonically isomorphic to A. [Centreof an abelian category is manifestly commutative and, in general, it gives us onlythe centre of the ring, that is Z(A). But here we are talking about honest schemesand hence, Z(A) = A].

ψ : A −→ End(IdA)

a 7−→ ψa such that ψa : IdA(M) −→ IdA(M) is just mult. by a ∀ M ∈Mod(A).

It is easy to see that the map is injective and a ring homomorphism. Now anyθ ∈ End(IdA) is a collection of endomorphisms θMM∈Mod(A) : M −→ M , suchthat for all M,N ∈Mod(A) and for all φ ∈ Hom(M,N),

MθM−−−−→ M

φ

yyφ

N −−−−→θN

N

the above diagram commutes. In particular, whenM = A, usingHomA(A,M) ∼=M via φ 7→ φ(1), we see that θN φ(1) = φ θA(1) = θA(1).φ(1). So θN = mult.by θA(1) ∈ A independent of N .

We can also get a derived analogue of the above result, which is, however, con-siderably weaker. Also it is claimed to be an easy consequence of the above theoremin [BO01].


Theorem 2.7. [Bondal Orlov [BO01]] Let X be a smooth irreducible projective va-riety with ample canonical or anti-canonical sheaf. If D = DbCoh(X) is equivalentas a graded category to DbCoh(X ′) for some other smooth algebraic variety X ′,then X is isomorphic to X ′.

Finally, consider a pre-additive category with a single object, say ∗. Then beinga pre-additive category Hom(∗, ∗) is endowed with an abelian group structure. Ifwe define a product on it by composition, then it is easy to verify that the twooperations satisfy the ring axioms. So Hom(∗, ∗) or simply End(∗) is a ring andthat is all we need to know about the pre-additive category. Extrapolating this lineof thought, we say that pre-additive categories generalize the concept of rings andsince schemes are concocted from commutative rings, it is reasonable to believe thatby some suitable constructions on something like a pre-additive category with somegeometric properties (= a Grothendieck category) we can find “non-commutativeschemes”.

A small discussion on construction of quotient categories

Recall that we call a full sub-category C of an abelian category A thick if thefollowing condition is satisfied:

for all short exact sequences in A of the form 0 −→M ′ −→M −→M ′′ −→ 0,we have

M ∈ C ⇐⇒ both M ′,M ′′ ∈ C

It becomes a Serre sub-category if it is coreflective (i.e., the canonical inclusionfunctor C −→ A has a right adjoint), besides being thick. This is essentially a “localterminology”.

Remark 2.8. What we call thick sub-categories out here are referred to as Serresub-categories by some authors. Further, note that in this definition, Serre sub-categories are themselves cocomplete (the image of the coproduct in the biggercategory under the adjoint functor of the inclusion belongs to the Serre sub-categoryand is easily seen to be a coproduct as well).

Now we construct the quotient of A by a thick sub-category C, denoted A/C, asfollows:

Ob(A/C) = objects of A.

HomA/C(M,N) = Lim−→

∀M′,N′ sub−obj.s.t.M/M′,N′∈C

HomA(M ′, N/N ′).

Of course, one needs to check that as M ′ and N ′ run through all sub-objects ofM and N respectively, such that N ′ and M/M ′ are in Ob(C), the abelian groupsHomC(M

′, N/N ′) form a directed system.The essence of this quotient construction is that the objects of C become isomor-

phic to zero.

Example 3. Let A = Mod(Z) and C = Torsion groups. Then one can show thatA/C ≃Mod(Q).


Let us define a functor from A/C to Mod(Q) by tensoring with Q. We simplifythe Hom sets of A/C. Using the structure theorem, write every abelian group asa direct sum of its torsion part and torsion-free part. If one of the variables istorsion, it can be shown that in the limit Hom becomes 0. So we may assume thatboth variables are torsion-free and for simplicity let us consider both of them to beZ. Then,

HomA/C(Z,Z) = HomA−→n

(nZ,Z)

=⋃

n

1

nZ

= Q = Hom(Q,Q)

This says that the functor is full, and an easy verification shows that it is faithfuland essentially surjective.

Let π denote the canonical quotient functor A −→ A/C. If π has a right adjointS : A/C −→ A called the section functor, then the thick sub-category C is calledlocalizing. It is known that in a Grothendieck category, the notions of a Serre sub-category and a localizing sub-category are the same. Further, if A is a Grothendieckcategory and C is a localizing sub-category, then A/C is again a Grothendieckcategory.

Spectrum of Grothendieck Categories

The passage between commutative rings and affine spaces is so smooth andtransparent that one gets easily tempted to directly generalize such ideas to thenon-commutative domain. For non-commutative rings which are finite modulesover their centres, there exists a satisfactory theory. Let A be a non-commutativering with centre, Z(A) = R. Then the pair (SpecR, A) is a nice geometrical object,

where A is the sheaf of algebras associated to the R-algebraA, satisfyingMod(A) ≃category of sheaves of A-modules. But many interesting algebras encountered inphysics fail to be centre-finite modules, for example, Weyl algebras. To associatea “space” to a non-commutative ring we work with not just the ring, but thecategory of modules over that ring. Don’t forget our paradigm - a “space” is justthe category of sheaves on it. But such a category is a Grothendieck category. So weare ready to enter the murky world of Grothendieck categories. The contents underthis sub-title are based entirely upon chapter 3 of A. Rosenberg’s book [Ros95] andfor details interested readers are requested to look it up (he talks about arbitraryabelian categories but we focus on Grothendieck categories). The aim is to concoctsomething resembling a scheme from a Grothendieck category. We shall discuss thedifferent attributes of a scheme very tersely one by one.

The underlying set (Spec(A))

Definition 2.9. A pre-order on Grothendieck Categories (≻):Fix a Grothendieck Category A. For any two objects X and Y of A we shall

write X ≻ Y if and only if Y is a subquotient of a finite coproduct of copies of Xi.e., there exists U such that the following holds


⊕finite

X ← U ։ Y

A pre-order is just reflexive and transitive. We introduce an equivalence and gomodulo that equivalence to bring in symmetry. Call X ∼ Y if and only if X ≻ Yand Y ≻ X . Then set |A| = ((Ob(A)/∼) , ≻). This is an ordered set.

Definition 2.10. Spectrum of A:Let S(A) = M ∈ Ob(A) |M 6= 0, ∀ 0 6= N → M, N ≻ M. The spectrum of

A, denoted Spec(A), is the ordered set of equivalence classes (with respect to ≻) ofelements of S(A).

Remark 2.11. It is obvious that simple objects (objects which have no non-zeroproper sub-objects) belong to S(A). It can also be shown that two simple objectsare equivalent (with respect to ≻) if and only if they are isomorphic. So in Spec(A)isomorphic simple objects are clubbed together.

Proposition 2.12. Let Q : A −→ B be an exact localization functor betweenGrothendieck categories. Then, for any P ∈ Spec(A), either Q(P ) = 0 or Q(P ) ∈Spec(B).

This proposition says that exact localizations almost respect spectrums (0 doesnot belong to the spectrum).

Realisation of Spec(A) in terms of Serre Sub-categories

While discussing quotient construction in categories we have already introducedthe notion of a Serre Sub-category.

A useful notation 〈M〉:For any M ∈ Ob(A) put

〈M〉 := full sub-category of A generated by objects N , such that N ⊁ M .

Lemma 2.13. For any M,M ′ ∈ Ob(A), M ≻ M ′ if and only if 〈M〉 ⊇ 〈M ′〉.Thus, M is equivalent to M ′ with respect to ≻ if and only if 〈M ′〉 = 〈M〉.

Let us continue to denote by 〈M〉 the equivalence class of 〈M〉. Then we havethe following map between posets.

(|A|,≻) −→ (〈M〉|M ∈ Ob(A),⊇)

M 7−→ 〈M〉

This is clearly one-to-one and hence, we have a realisation of |A| in terms ofcertain sub-categories of A.

Proposition 2.14. If an object P of the category A belongs to Spec(A), then 〈P 〉is a Serre sub-category of A.

Sketch of Proof:Let P ∈ Spec(A). We need to show that for any exact sequence 0 −→ M ′ −→

M −→M ′′ −→ 0, M ∈ 〈P 〉 if and only if both M ′ ∈ 〈P 〉 and M ′′ ∈ 〈P 〉. Considerthe following diagram,


0

↑

⊕finite

M ′′

↑i

տ

⊕finite

Mj← K ։ P

↑ ւk

⊕finite

M ′

↑

0

where i, j and k are supposed to be injections. Then existences of i and k easilyimply the existence of j (just compose k with the map ⊕

finiteM ′ −→ ⊕

finiteM). Now

suppose j exists. Then, ⊕finite

M ′ ← K ∩ ( ⊕finite

M ′) −→ P =⇒ M ′ ≻ L → P and

⊕finite

M ′′ ← K ′ −→ P =⇒ M ′′ ≻ L′ → P , where K ′ = K/(K ∩ ( ⊕finite

M ′)). But

as P ∈ Spec(A), L,L′ sub-objects of P ⇒ L,L′ ≻ P . Hence, both M ′,M ′′ ≻ P .We have actually proved, M /∈ 〈P 〉 ⇐⇒ both M ′,M ′′ /∈ 〈P 〉.

For coreflectiveness, we need a right adjoint for ı : 〈P 〉 −→ A. For any M ∈Ob(A), let 〈P 〉(M) denote the set of all sub-objects ofM which belong to 〈P 〉. SinceA is a Grothendieck category it has the (sup) property. So M := sup(〈P 〉(M))exists in A. But, as 〈P 〉 contains all sub-objects and quotient-objects of its objects,M ∈ Ob(〈P 〉). Define the adjoint of ı, denoted ı!, by

ı! : A −→ 〈P 〉

M 7−→ M

It requires some more work to prove the following proposition.

Proposition 2.15. For any object M of A, such that 〈M〉 is a Serre sub-category,there is an object P ∈ Spec(A) which is equivalent to M , i.e., 〈M〉 = 〈P 〉.

Putting two [2.14] and two [2.15] together we get a realisation of Spec(A).

Spec(A) = 〈M〉 | M ∈ Ob(A) and 〈M〉 is a Serre sub-category

Categorical incarnation of Local rings of Spec(A)

As we have just seen, to every object M ∈ Spec(A) one can associate a localizingsub-category 〈M〉 and hence an exact localization QM : A −→ A/〈M〉.

A non-zero object M of A is called quasi-final if N ≻M for any non-zero objectof A or equivalently 〈M〉 is the zero sub-category.

A quasi-final object, if it exists, clearly belongs to the spectrum and any twosuch objects are equivalent.


Definition 2.16. Local abelian category An abelian category A is called localif it has a quasi-final object.

Now we have the following proposition to justify the christening.

Proposition 2.17. Let A be a local abelian category. Then the center of A is acommutative local ring.

Though we have introduced the notion of locality for arbitrary abelian categories,we go back to our favourite Grothendieck categories. In a usual (commutative)scheme the stalk of the structure sheaf at a point is a local ring. So far we haveobtained a set, which is Spec(A), and if we have done something sane then forevery point M ∈ Spec(A) we should expect to get a “local object”. Coupled with[2.17] the following proposition gives us precisely what we expect.

Proposition 2.18. For any object M ∈ Spec(A), the quotient category A/〈M〉 islocal.

Sketch of proof:Let Q : A −→ A/〈M〉 be the exact localization functor. We shall show that

Q(M) is quasi-final and hence, A/〈M〉 is local. Take any non-zero object X ∈A/〈M〉. X can also be thought of as an object in A [refer to quotient construction]and since it is a non-zero object in the quotient category, X /∈ Ob(〈M〉). Thisimplies that X ≻M . Since Q, being an exact functor, respects ≻, we have Q(X) =X ≻ Q(M). Since X was chosen arbitrarily, Q(M) is a quasi-final object.

Categorical incarnation of the residue field at a point

As is wont, we need to beat around the bush a little bit. Some more technicaljargon have to show up.

Definition 2.19. Support of an object:The support of M ∈ Ob(A), denoted Supp(M) is the set of all 〈P 〉 ∈ Spec(A)

such that M ≻ P . Or, in other words,

Supp(M) = 〈P 〉 ∈ Spec(A)|Q〈P 〉M 6= 0

For any subset W ⊆ Spec(A), let A(W ) be the full sub-category generated byall objects M of the category A such that Supp(M) ⊆W .

Lemma 2.20. A(W ) = ∩P∈W⊥

〈P 〉, where W⊥ := Spec(A)−W .

Sketch of proof:Let M ∈ Ob( ∩

P∈W⊥

〈P 〉) ⇐⇒ Supp(M)∩Ob( ∩P∈W⊥

〈P 〉) = ∅ ⇐⇒ Supp(M) ⊆W

i.e., M ∈ Ob(A(W )).

Now if one is ready to believe that the intersection of any set of Serre sub-categories is once again a Serre sub-category, we may conclude from the abovelemma that A(W ) is a Serre sub-category. For any point 〈P 〉 ∈ Spec(A) considerthe category

K(P ) := A(Supp(P ))/〈P 〉


Now we denote by K(P ) the full sub-category of K(P ) generated by all objectsM of K(P ) which are supremums of their sub-objects V, which satisfy 〈V 〉 = 〈P 〉.

We call the category K(P ) the residue category of 〈P 〉. One can check that it islocal and its spectrum is singleton.

The Zariski topology on Spec(A)

To define the analogue of Zariski topology we need some ingredients first.

Definition 2.21. Topologizing sub-category:A full sub-category C of A is called topologizing if for any exact sequence in A

0 −→M ′ −→M −→M ′′ −→ 0

M ∈ C =⇒ both M ′,M ′′ ∈ C. Besides, it should also be closed under finitecoproducts.

Clearly any thick sub-category is topologizing, and so is any Serre sub-category(Serre =⇒ thick).

Definition 2.22. Gabriel product of topologizing sub-categories:For any two topologizing sub-categories S, T of A define the Gabriel product S•T

as the full sub-category of A generated by all objects M of A such that there existsan exact sequence in A

0 −→M ′ −→M −→M ′′ −→ 0

with M ′ ∈ Ob(T ) and M ′′ ∈ Ob(S).

One can check that the Gabriel product of two topologizing sub-categories isalso a topologizing one. Further,

S • (T • U) = (S • T ) • U and 0 • S = S = S • 0

It is also evident from the definitions that a topologizing sub-category T is thickif and only if T • T = T .

Definition 2.23. Closed and left closed sub-categories:A topologizing sub-category is closed (resp. left closed) if is coreflective (resp.

reflective) as well. [coreflective (resp. reflective) means that the canonical inclusionfunctor has a right (resp. left) adjoint].

It needs some formal arguments to prove that the Gabriel product of two closed(resp. left closed) sub-categories is also closed (resp. left closed). Left closednessis also preserved under arbitrary intersections. Now we define for any topologizingsub-category (in particular, for any left closed sub-category)

V (T ) := 〈P 〉 ∈ Spec(A)|P ∈ Ob(T )This notation is not without any foresight. Now we are ready to define the

Zariski Topology, denoted ZT .

Definition 2.24. Zariski Topology (ZT ):It is the topology generated by the closed sets of the form V (T ), where T runs

through the set of all left closed sub-categories of A.

The fact that the sets of the form V (T ) play the role of the closed subsets ofZT is corroborated by the following proposition.


Proposition 2.25. The set ZT is closed under finite unions and arbitrary inter-sections.

Sketch of proof:(a) Union: V (S) ∪ V (T ) = V (S • T ).Clearly S ⊆ S • T ⊇ T . which implies the inclusion V (S) ∪ V (T ) ⊆ V (S • T ).For the other inclusion let us take any 〈P 〉 ∈ V (S •T ) i.e., P ∈ Spec(A)∪Ob(S •

T ). The latter means that there exists an exact sequence

0 −→ P ′ −→ P −→ P ′′ −→ 0

in which P ′ ∈ Ob(T ) and P ′′ ∈ Ob(S).If P ′ 6= 0, then P ′ ≻ P =⇒ P ∈ Ob(T ).If P ′ = 0, then P ≃ P ′′ =⇒ P ∈ Ob(S).

Therefore, we have

Spec(A) ∪Ob(S • T ) ⊆ (Spec(A) ∪Ob(S)) ∪ (Spec(A) ∪Ob(T ))

which implies the desired inclusion.

(b) Intersection:Let Ω be any set of left closed sub-categories of A. Clearly, ∩

T ∈ΩV (T ) =

V ( ∩T ∈ΩT ). But left closed sub-categories, as stated earlier, are closed under ar-

bitrary intersections. So ∩T ∈ΩT is also left closed =⇒ V ( ∩

T ∈ΩT ) ∈ ZT .

Remark 2.26. It has been shown in the book [Ros95] that (Spec(A),ZT ) is quasi-compact as a topological space if the generator of the Grothendieck Category A isof finite type (an object M of A is said to be of finite type if, for any directed set Ωof sub-objects of M such that supΩ = M , there already exists a sub-object M ′ ∈ Ωsuch that M ′ = M). Further, for an associative unital ring R, (Spec(Mod(R)),ZT )is quasi-compact and has a basis of quasi-compact open subsets. It should also bebourne in mind that the Zariski topology is not satisfactory in all circumstances.There are some other canonical topologies, denoted τ∗ and τ∗, for which interestedreaders may refer to [Ros95] and [Ros98].

Sheaves on Spec(A)

We already know that the closed sets of ZT are of the form V (T ). Taking thecomplements in Spec(A) we get the open sets, which we denote by Open(ZT ) (thiscan easily be thought of as a category with inclusions being the only morphisms).To any U ∈ Open(ZT ) we assign the Serre sub-category 〈U〉 := ∩

P∈U〈P 〉. According

to lemma [2.20] 〈U〉 = A(U⊥), where U⊥ is just the complementary closed subsetof U in ZT .

The “structure sheaf” is obtained as a sheaf of commutative rings via the functorOA.

OA : Open(ZT )op −→ (Comm. rings)

U 7−→ Z(A/〈U〉)

where Z(A/〈U〉) stands for the centre of the category A/〈U〉.


To any M ∈ Ob(A) we associate a functor M . Let QA/〈U〉 : A −→ A/〈U〉 and putMU = QA/〈U〉(M).

M : Open(ZT )op −→Ab

U 7−→MU

(in an ab. cat. every object is an abelian group object.)

The M , so defined, is called the sheaf associated to the object M . Of course, onehas to check that both OA and M satisfy the sheaf conditions. Some more workactually shows that M defines a sheaf of OA-modules (θ : IdA/〈U〉 −→ IdA/〈U〉then θ is easily seen to act on MU ). Further, one has Mod(OA(U)) ≃ A/〈U〉.

In the book [Ros95] Rosenberg also talks about relative spectra associated tofunctors F : A −→ B, but due to lack of time and space we skip those details.

A glaring omission:A. Rosenberg has argued in his book [Ros95] that his spectrum is actually smaller

than the other prevalent ones and is also easier to compute. As an illustration he hascomputed the spectra of so-called hyperbolic rings i.e., rings of the form R〈X,Y 〉subject to the relations Xr = θ(r)X , Y r = θ−1(r)Y , XY = u and Y X = θ−1(u),where θ ∈ Aut(R),r ∈ R and u ∈ Z(R). Such a spectrum, when specialized to amodule category over an associative unital ring R, coincides with the so-called leftspectrum of R.

Conclusion:We have pushed a heavy ball up to the top of a hill and now we can just let

it roll down via the reconstruction theorem [2.6] to get back to the square posi-tion. To be precise, suppose A = Qcoh(X), where X is a noetherian (actuallyquasi-compact and quasi-separated is enough) scheme. Then it has been proved in[Ros98] that the ringed space (Spec(A),OA) is isomorphic to (X,OX). Actuallythe reconstruction works for arbitrary schemes. We have deliberately worked overGrothendieck categories to “conform to the title of the section”. Unfortunately, thecategory of quasi-coherent sheaves may not be a Grothendieck category for arbi-trary schemes, but for noetherian ones, it is. For a noetherian scheme X , we areallowed to concentrate on A = Coh(X) by [2.5] and the comment following that.Then the reconstruction maps are (in less than a nutshell)

φ : X −→Spec(A)

x 7−→〈Px〉 the sky-scraper sheaf,

whose only non-zero stalk in the residue field at x.

ψ : Spec(A) −→X

〈M〉 7−→the generic point of the support of M .

〈M〉 ∈ Spec(A) =⇒ Supp(M) is irreducible and closed in X .

One has to check that ψ φ = IdX and φ ψ = IdSpec(A) and also that they areindeed morphisms of ringed spaces. The definition of Spec(A) is such that sheaveswith reducible supports cannot belong to it. This is one of the interesting pointsof the proof.


This excerpt is in the spirit of non-commutative algebraic geometry subsumingrepresentation theory. The classical problem of determining all irreducible represen-tations of an algebra R (typically enveloping algebras) can be reduced to the studyof the spectrum of the category, Mod(R). We have provided a general recipe toconstruct ringed spaces from Grothendieck categories and Mod(R) is an archetypeof such categories. The existence of a generator in a Grothendieck category is quitea strong hypothesis as we know virtually everything about the category from themodule category of the endomorphism ring of the generator (Gabriel-Popescu [2.4]).This non-commutative spectral theory reveals important classes of irreducible rep-resentations which are, otherwise, out of reach of the classical theory (Verma andHarish-Chandra modules).


3. Non-commutative Projective Geometry

Although so far we have done everything in a very general set-up, we would soonlike to concentrate on, what is called, non-commutative projective geometry. Butbefore that let us go through one nice result in the affine case. Let X be an affinescheme and put A = Γ(X,OX). Then it is classical that Qcoh(X) is equivalentto Mod(A). This immediately begs the question of which Grothendieck categoriescan be written as Mod(A) for some possibly non-commutative ring A. The answeris given by the theorem below.

Theorem 3.1. [Ste75] Let C be a Grothendieck category with a projective generatorG and assume that G is small [i.e., Hom(G,−) commutes with all direct sums].Then C ≃Mod(Aop), for A = End(G).

Note that the Gabriel Popescu Theorem [2.4] gave just a fully faithful embeddingwith an exact left adjoint and not an equivalence.

Now we are ready to discuss a model of non-commutative projective geometryafter Artin and Zhang [AZ94]. However, we would also like to bring into the noticeof the readers the works done by Verevkin (see [Ver92]). We have invested enoughtime in trying to convince ourselves that the categorical language should be adopted.We shall make no exception to that leitmotif here, but the rules of the game willbe changed slightly. Fix an algebraically closed field k; then all categories willbe assumed to be k-linear abelian categories [i.e., the bifunctor Hom ends up inMod(k)]. Since in commutative algebraic geometry one mostly deals with finitelygenerated k-algebras, which are noetherian, here we assume that our k-algebras areat least right noetherian. Let R be a graded algebra. Then we introduce some morecategories (which are all k-linear):

Gr(R) := category of Z-graded right R-modules, with degree 0 morphisms.

Tor(R) := full subcategory of Gr(R) generated by torsion modules

(i.e., M such that ∀ x ∈M , xR>s = 0 for some s), which is thick.

QGr(R) := the quotient category Gr(R)/Tor(R) (refer to the sub-section on

quotient construction in the previous chapter).

Remark 3.2. Standard Convention. If XY uvw(. . . ) denotes an abelian category,then we shall denote by xyuvw(. . . ) the full sub-category consisting of noetherianobjects and if A,B, . . . ,M,N, . . . denote objects in Gr(R) then we shall denote byA,B, . . . ,M,N , . . . the corresponding objects in QGr(R).

Some people denote QGr(R) by Tails(R), but we shall stick to our notation.We denote the quotient functor Gr(R) −→ QGr(R) by π. It has a right adjointfunctor ω : QGr(R) −→ Gr(R) and so, for all M ∈ Gr(R) and F ∈ QGr(R) oneobtains

HomQGr(R)(πM,F ) ∼= HomGr(R)(N,ωF ).

The Hom’s of QGr(R) take a more intelligible form with the assumptions on R.It turns out that for any N ∈ gr(R) and M ∈ Gr(R)


HomQGr(R)(πN, πM) ∼= Lim→

HomGr(R)(N>n,M)(22)

For any functor F from a k-linear category C equipped with an autoequivalences, we denote by F the graded analogue of F given by F (A) := ⊕

n∈Z

F (snA) for any

A ∈ Ob(C). Further, to simplify notation we shall sometimes denote snA by A[n]when there is no chance of a confusion.

Keeping in mind the notations introduced above we have

Lemma 3.3. ωπM ∼= Lim→

HomR(R>n,M)

Sketch of proof:

ωπM = HomR(R,ωπM) [since R ∈ gr(R)]

= ⊕d∈Z

HomGr(R)(R,ωπM [d])

= ⊕d∈Z

HomQGr(R)(πR, πM [d]) [by adjointness of π and ω]

= ⊕d∈Z

Lim→

HomGr(R)(R>n,M [d]) [by [22]]

= Lim→⊕d∈Z

HomGr(R)(R>n,M [d])

= Lim→

HomR(R>n,M)

The upshot of this lemma is that, there is a natural equivalence of functorsω ≃ Hom(A,−).

Proj R

Let X be a projective scheme with a line bundle L . Then the homogeneouscoordinate ring B associated to (X,L ) is defined by the formula B = ⊕

n∈N

Γ(X,L n)

with the obvious multiplication. Similarly, if M is a quasi-coherent sheaf on X ,Γh(M ) = ⊕

n∈N

Γ(X,M ⊗ L n) defines a graded B-module. Thus, the compostion

of Γh with the natural projection from Gr(B) to QGr(B) yields a functor Γh :Qcoh(X) −→ QGr(B). This functor works particularly well when L is ample asis evident from the following fundamental result due to Serre.

Theorem 3.4. [Ser55] 1. Let L be an ample line bundle on a projective schemeX. Then the functor Γh(−) defines an equivalence of categories between Qcoh(X)and QGr(B).

2. Conversely, if R is a commutative connected graded k-algebra, that is R0 = kand it is generated by R1 as an R0-algebra, then there exists a line bundle L overX = Proj(R) such that R = B(X,L ), up to a finite dimensional vector space.Once again, QGr(R) ≃ Qcoh(X).

In commutative algebraic geometry one defines the Proj of a graded ring to bethe set of all homogeneous prime ideals which do not contain the augmentationideal. This notion is not practicable over arbitrary algebras. However, Serre’stheorem filters out the essential ingredients to define the Proj of an arbitrary


algebra. The equivalence is controlled by the category Qcoh(X), the structuresheaf OX and the autoequivalence given by tensoring with L , which alludes to thepolarization of X . Borrowing this idea we get to the definition of Proj. Actuallyone should have worked with a Z-graded algebra R and defined its Proj but it hasbeen shown in [AZ94] that, with the definition to be provided below, Proj R is thesame as Proj R>0. Hence, we assume that R is an N-graded k-algebra. Gr(R)has a shift operator s such that s(M) = M [1] and a special object, RR. We canactually recover R from the assortment (Gr(R), RR, s) by

R = ⊕i∈N

Hom(RR, si(RR))

and the composition is given as follows: a ∈ Ri and b ∈ Rj , then ab = sj(a) b ∈Ri+j .

Let R denote the image of R in QGr(R) and we continue to denote by s theautoequivalence induced by s on QGr(R).

Definition 3.5. (Proj R)The triple (QGr(R),R, s) is called the projective scheme of R and is denoted

Proj R. Keeping in mind our convention we denote (qgr(R),R, s) by proj R. Thisis also equally good due to [2.5].

Characterization of Proj R

Having transformed Serre’s theorem into a definiton, it is time to address themost natural question - which triples (C,A, s) are of the form Proj(R) for somegraded algebra R? This problem of characterization has been dealt with compre-hensively by Artin and Zhang. We would be content by just taking a quick look atthe important points. Let us just acquaint ourselves with morphisms of such triples.A morphism between (C,A, s) and (C′,A′, s′) is given by a triple (f, θ, µ), wheref : C −→ C′ is a k-linear functor, θ : f(A) −→ A′ is an isomorphism in C′ and µ isa natural isomorphism of functors f s −→ s′ f . The question of characterizationis easier to deal with when s is actually an automorphism of C. To circumvent thisproblem, an elegant construction has been provided in [AZ94] whereby one can passon to a different triple, where s becomes necessarily an automorphism. If s is anautomorphism one can take negative powers of s as well and it becomes easier todefine the graded analogues of all functors (refer to [3.2]). Sweeping that discussionunder the carpet, henceforth, we tacitly assume that s is an automorphism of C(even though we may write s to be an autoequivalence).

The definition of Proj was conjured up from Serre’s theorem where the triplewas (Qcoh(X),OX ,− ⊗L ). Of course, one can easily associate a graded algebrato (C,A, s).

Γh(C,A, s) = ⊕n>0

Hom(A, snA)

with multiplication a.b = sn(a)b for a ∈ Hom(A, smA) and b ∈ Hom(A, snA).But L has to be ample and we need a notion of ampleness in the categorical

set-up.

Definition 3.6. (Ampleness)


Assume that C is locally noetherian. Let A ∈ Ob(C) be a noetherian object andlet s be an autoequivalence of C. Then the pair (A, s) is called ample if the followingconditions hold:

1. For every noetherian object O ∈ Ob(C) there are positive integers l1, . . . , lp

and an epimorphism fromp⊕i=0A(−li) to O.

2. For every epimorphism between noetherian objects P −→ Q the induced mapHom(A(−n),P) −→ Hom(A(−n),Q) is surjective for n≫ 0.

Remark 3.7. The first part of this definition corresponds to the standard definitionof an ample sheaf and the second part, to the homological one.

Now we are in good shape to state one part of the theorem of Artin and Zhangwhich generalizes that of Serre.

Theorem 3.8. Let (C,A, s) be a triple as above such that the following conditionshold:

(H1) A is noetherian,(H2) A := Hom(A,A) is a right noetherian ring and Hom(A,M) is a finite

A-module for all noetherian M, and(H3) (A, s) is ample.Then C ≃ QGr(B) for B = Γh(C,A, s). Besides, B is right noetherian.

The converse to this theorem requires an extra hypothesis, which is the so-calledχ1 condition. One could suspect, and rightly so, that there is a χn condition forevery n. They are all some kind of condition on the graded Ext functor. However,they all look quite mysterious at a first glance. Actually most naturally occurringalgebras satisfy them but the reason behind their occurrence is not well understood.We shall discuss them in some cases later but we state a small proposition first.

Proposition 3.9. Let M ∈ Gr(B) and fix i > 0. There is a right B-modulestructure on ExtnB(B/B+,M) coming from the right B-module structure of B/B+.Then the following are equivalent:

1. for all j 6 i, ExtjB(B/B+,M) is a finite B-module;

2. for all j 6 i, ExtjB(B/B>n,M) is finite for all n;

3. for all j 6 i and all N ∈ Gr(B), ExtjB(N/N>n,M) has a right boundindependent of n;

4. for all j 6 i and all N ∈ Gr(B), Lim→

ExtjB(N/N>n,M) is right bounded.

The proof is a matter of unwinding the definitions of the terms suitably and thenplaying with them. We shall do something smarter instead - make a definition outof it.

Definition 3.10. (χ conditions)A graded algebra B satisfies χn if, for any finitely generated graded B-module

M , one of the equivalent conditions of the above proposition is satisfied (after sub-stituting i = n in them). Moreover, we say that B satisfies χ if it satisfies χn forevery n.

Remark 3.11. Since B/B+ is a finitely generated B0-module (B0 = k) we couldhave equally well required the finiteness of ExtnB(B/B+,M) over B0 = k i.e.,dimkExt

nB(B/B+,M) <∞ for χn.


Let B be an N-graded right noetherian algebra and π : Gr(B) −→ QGr(B).

Theorem 3.12. If B satisfies χ1 as well, then (H1), (H2) and (H3) hold for thetriple (qgr(B), πB, s). Moreover, if A = Γh(QGr(B), πB, s), then Proj B is iso-morphic to Proj A via a canonical homomorphism B −→ A. [We have a canonicalmap Bn = HomB(B,B[n]) −→ Hom(πB, πB[n]) = An given by the functor π.]

The proofs of the these theorems are once again quite long and involved. So theyare left out. What we need now is a good cohomology theory for studying suchnon-commutative projective schemes.

Cohomology of Proj R

The following rather edifying theorem due to Serre gives us some insight intothe cohomology of projective (commutative) spaces.

Theorem 3.13. [Har77] Let X be a projective scheme over a noetherian ring A,and let OX(1) be a very ample invertible sheaf on X over SpecA. Let F be acoherent sheaf on X. Then:

1. for each i > 0, Hi(X,F ) is a finitely generated A-module,2. there is an integer n0, depending on F , such that for each i > 0 and each

n > n0, Hi(X,F (n)) = 0.

There is an analogue of the above result and we zero in on that. We have alreadycome across the χ conditions, which have many desirable consequences. Actuallythe categorical notion of ampleness doesn’t quite suffice. For the desired result to gothrough, we need the algebra to satisfy χ too. Without inundating our minds withall the details of χ we propose to get to the point i.e., cohomology. Set πR = R.On a projective (commutative) scheme X one can define the sheaf cohomology ofF ∈ Coh(X) as the right derived functor of the global sections functor i.e., Γ. ButΓ(X,F ) ∼= HomOX (OX ,F ). Buoyed by this fact and having faith in the peopleworking in “motivic” areas, who dream of constructing a universal cohomologytheory for varieties (see [SV00]) via the Ext groups, we try to define the cohomologyfor everyM ∈ qgr(R) as

Hn(M) := ExtnR(R,M)

However, taking into consideration the graded nature of our objects we alsodefine the following:

Hn(Proj R,M) := ExtnR(R,M) = ⊕i∈Z

ExtnR(R,M[i])

The categoryQGr(R) has enough injectives and one can choose a nice “minimal”injective resolution of M to compute its cohomologies, the details of which areavailable in the chapter 7 of [AZ94].

Let M ∈ Gr(R) and write M = πM . Then one should observe that

Hn(M) = ExtnQGr(R)(R,M)

∼= Lim→

ExtnGr(R)(R>n,M) [by[22]](23)

As R-modules we have the following exact sequence,


0 −→ R>n −→ R −→ R/R>n −→ 0

For any M ∈ Gr(R), the associated Ext sequence in Gr(R) looks like

. . . Extj(R/R>n,M) −→ Extj(R,M) −→ Extj(R>n,M) −→ . . .

Since R is projective as an R module, Extj(R,M) = 0 for every j > 1. Thus,we get the following exact sequence

0→ Hom(R/R>n,M)→M → Hom(R>n,M)→ Ext1(R/R>n,M)→ 0(24)

and, for every j > 1, an isomorphism

Extj(R>n,M) ∼= Extj+1(R/R>n,M)(25)

The following theorem is an apt culmination of all our efforts.

Theorem 3.14. (Serre’s finiteness theorem)Let R be a right noetherian N-graded algebra satisfying χ, and let F ∈ qgr(R).

Then,(H4) for every j > 0, Hj(F ) is a finite right R0-module, and

(H5) for every j > 1, Hj(F ) is right bounded; i.e., for d≫ 0, Hj(F [d]) = 0.

Sketch of proof:Write F = πM for some M ∈ gr(R). Suppose that j = 0. Since χ1(M) holds,

ExtiR(R/R>n,M) is a finite R-module for each i = 1, 2 and together with [24] it

implies that ωF ∼= H0(F ) is finite (recall ω from [3.2]). Now taking the 0-gradedpart on both sides we get (1) for j = 0.

Suppose that j > 1. Since R satisfies χj+1, invoking proposition [3.9] we get

Lim→

Extj+1R (R/R>n,M)

is right bounded. Combining [23] and [25] this equals Hj(F ). This immediately

proves (2) as Hj(F )d = Hj(F [d]). We now need left boundedness and local

finiteness of Hj(F ) to finish the proof of (1) for j > 1. These we have alreadyobserved (at least tacitly) but one can verify them by writing down a resolution ofR/R>n involving finite sums of shifts of R, and then realizing the cohomologies as

sub-quotients of a complex of modules of the form HomR(p⊕i=0R[li],M).

Our discussion does not quite look complete unless we investigate the questionof the “dimension” of the objects that we have defined.

Dimension of Proj R

The cohomological dimension of Proj R denoted by cd(Proj R) is defined to be

cd(Proj R) :=

supi | Hi(M) 6= 0 for someM ∈ qgr(R) if it is finite,

∞ otherwise.


Remark 3.15. As Hi commutes with direct limits one could have used QGr(R) inthe definition of the cohomological dimension.

The following proposition gives us what we expect from a Proj constructionregarding dimension and also provides a useful way of calculating it.

Proposition 3.16. 1. If cd(Proj R) is finite, then it is equal to supi | Hi(R) 6=0.

2. If the left global dimension of R is d <∞, then cd(Proj R) 6 d− 1.

Sketch of proof:1. Let d be the cohomological dimension of Proj R. It is obvious that supi | Hi(R) 6=

0 6 d. We need to prove the other inequality. So we choose an object for whichthe supremum is attained i.e., M ∈ qgr(R) such that Hd(M) 6= 0 and, hence,

Hd(M) 6= 0. By the ampleness condition we may write down the following exactsequence

0 −→ N −→p⊕i=0R[−li] −→M −→ 0

for some N ∈ qgr(R). By the long exact sequence of derived functors Hi wehave

. . . −→p⊕i=0Hd(R[−li]) −→ Hd(M) −→ Hd+1(N ) = 0

This says that Hd(R[−li]) 6= 0 for some i and hence, Hd(R) 6= 0.

2. It has already been observed that Hi(M) ∼= Limn→∞

Exti(R>n,M) for all i > 0.

Now, if the left global dimension of R is d, then Extj(N,M) = 0 for all j > d and

all N,M ∈ Gr(R). Putting N = R/R> and using [25] we get Hd(M) = 0 for allM ∈ Gr(R). Therefore, cd(Proj R) 6 d− 1.

Remark 3.17. If R is a noetherian AS-regular graded algebra, then the Gorensteincondition can be used to prove that cd(Proj R) is actually equal to d− 1.

Some Examples (mostly borrowed from [AZ94])

Example 4. (Twisted graded rings)Let σ be an automorphism of a graded algebra A. Then we define a new multi-

plication ∗ on the underlying graded k-module A = ⊕nAn by

a ∗ b = aσn(b)

where a and b are homogeneous elements in A and deg(a) = n. Then algebrais called the twist of A by σ and it is denoted by Aσ. By [TVdB91] and [Zha96]gr(A) ≃ gr(Aσ) and hence, proj(A) ≃ proj(Aσ).

For example, if A = k[x, y] where deg(x) = deg(y) = 1, then any linear operatoron the space A1 defines an automorphism, and hence, a twist of A. If k is analgebraically closed field then, after a suitable linear change of variables, a twist canbe brought into one of the forms kq[x, y] := kx, y/(yx− qxy) form some q ∈ k, or


kj [x, y] := kx, y/(x2+xy−yx). Hence, proj k[x, y] ≃ proj kq[x, y] ≃ proj kj [x, y].The projective scheme associated to any one of these algebras is the projective lineP1.

Example 5. (Changing the structure sheaf)Though the structure sheaf is a part of the definition of Proj, one might ask,

given a k-linear abelian category C, which objects A could serve the purpose of thestructure sheaf. In other words, for which A do the conditions (H1), (H2) and(H3) of Theorem [3.8] hold? Since (H3) involves both the structure sheaf and thepolarization s, the answer may depend on s. We propose to illustrate the possibilitiesby the simple example in which C = Mod(R) when R = k1 ⊕ k2, where ki = k fori = 1, 2 and where s is the automorphism which interchanges the two factors. Theobjects of C have the form V ≃ kn1

1 ⊕ kn2

2 , and the only requirement for (H1), (H2)and (H3) is that both r1 and r2 be not zero simultaneously.

We have sn(V ) = kr21 ⊕ kr12 if n is odd and sn(V ) = V otherwise. Thus, ifwe set A = V and A = Γh(C,A, s), then An ≃ kr1×r11 ⊕ kr2×r22 if n is even, andAn ≃ k

r1×r21 ⊕ kr2×r12 otherwise. For example, if r1 = 1 and r2 = 0, then A ≃ k[y],

where y is an element of degree 2. Both of the integers ri would need to be positiveif s were the identity functor.

Example 6. (Commutative noetherian algebras satisfy χ)Let A be a commutative noetherian k-algebra. Then the module structure on

ExtnA(A/A+,M) can be obtained both from the right A-module structure of A/A+

and that of M . Choose a free resolution of A/A+, consisting of finitely generatedfree modules. The cohomology of this complex of finitely generated A-modules isgiven by the Ext’s, whence they are finite.

Example 7. (Noetherian AS-regular algebras satisfy χ)If A is a noetherian connected N-graded algebra having global dimension 1, then A

is isomorphic to k[x], where deg(x) = n for some n > 0, which satisfies χ by virtueof the previous example. In higher dimensions we have the following proposition.

Proposition 3.18. Let A be a noetherian AS-regular graded algebra of dimensiond > 2 over a field k. Then A satisfies the condition χ.

Sketch of proof:A is noetherian and locally finite (due to finite GKdim). For such an A it is

easy to check that Extj(N,M) is a locally finite k-module whenever N,M are finite.

A0 is finite and hence, Extj(A0,M) is locally finite for every finite M and every j.Since A is connected graded A0 = k. For any n and any finite A-module M we firstshow that Extn(A0,M) = Extn(k,M) is bounded using induction on the projective

dimension of M . If pd(M) = 0, then M =p⊕i=0A[−li]. By the Gorenstein condition

(see definition [1.3]) of AS-regular algebra A, Extn(k,A[−li]) is bounded for eachi. Therefore, so is Extn(k,M). If pd(M) > 0, we choose an exact sequence

0 −→ N −→ P −→M −→ 0

where P is projective. Then pd(N) = pd(M) − 1. By induction, Extn(k,N)and Extn(k, P ) are bounded, hence, so is Extn(k,M). Now A/A+ is finite andwe have just shown Extn(k,M) is finite (since bounded together with locally finiteimplies finite); then Hom(A/A+, Ext

n(k,M)) ∼= Extn(A/A+,M) is locally finite


and clearly bounded. Therefore, Extn(A/A+,M) is finite for every n and everyfinite M .

Remark 3.19. In particular the AS-regular algebras of dimension 3 (they are auto-matically noetherian), which we have seen in chapter 1, satisfy χ. Also it shouldbe mentioned that all noetherian N-graded PI algebras satisfy χ.

There is a host of other examples on algebras satisfying χ up to varying degreesfor which we refer the interested readers to [AZ94], [Rog02] and [SZ94a].

Non-commutative smooth proper varieties

The following excerpt is just a leaf out of the article [SVdB01]. Instead of tryingto define everything starting from a graded algebra, we now endeavour to definethings abstractly and then obtain a description of them in terms of some suitablegraded algebras. We first focus on the homological properties that smooth propervarieties have.

Let X be a smooth, proper and connected k-scheme of dimension n and letC = Coh(X). Then the following properties stare at us discernibly.

(C1) C is noetherian.(C2) C is Ext-finite [i.e., Exti(A,B) <∞ for all A,B ∈ Ob(C) and all i].(C3) C has homological dimension n.

Unfortunately, easily obtained conditions are not always sufficient and we needto look for more. A fundamental, but slightly more subtle, property of smoothproper schemes is the Serre Duality Theorem - there exists a dualizing sheaf ω suchthat for every F ∈ C there are natural isomorphisms

Hi(X,F ) ∼= Extn−i(F , ω)∗ [where ∗ refers to k-dual].

Bondal and Kapranov have added to our arsenal a very elegant reformulation ofSerre duality [BK89]. Let Db(G) denote the derived category of bounded complexesover an abelian k-linear category G.

Definition 3.20. (Serre functor)A Serre functor on Db(G) is an autoequivalence F : Db(G) −→ Db(G) such that

there are bifunctorial isomorphisms

Hom(A,B) ∼= Hom(B,FA)∗(26)

which are natural for all A,B ∈ Ob(Db(G)).

Then it is shown that Serre duality can be reinterpreted as saying:

(C4) C satisfies Serre duality in the sense that there exists a Serre functor onDb(C).

We adapt our condition (C2) to the graded scenario and demand Ext-finiteness,instead of Ext-finiteness. Classification of abelian k-linear categories satisfying(C1),(C2),(C3) and (C4) is a tough nut to crack at the moment. However, forhereditary categories, i.e., the ones of homological dimension one, the situation isslightly more tractable.


The most obvious hereditary noetherian Ext-finite abelian k-linear categoriessatisfying Serre duality in the form of (C4) are:

(E1) Let X be a smooth projective connected curve over k with function field Kand let O be a sheaf of hereditary OX orders in Mn(K) (see [Rei75]). Then oneproves exactly as in the commutative case that C1 = Coh(O) satisfies Serre duality.The Serre functor is given by tensoring with Hom(O, ωX)[1].

We now come to three, rather special, examples (all from [SVdB01]) for which(C1)-(C4) need some work to be verified.

(E2) C2 consists of the finite dimensional nilpotent representations of the quiver

An [Rin84] (possibly with n = ∞) with all arrows oriented in the same direction.The Serre functor is given by rotating one place in the direction of the arrows andshifting one place to the left in the derived category.

(E3) C3 = qgrZ/2Z(k[x, y]) consists of the finitely generated Z/2Z-graded k[x, y]-modules modulo the finite dimensional ones. The Serre functor is given by the shiftM 7−→M(−1,−1)[1]

(E4) The category C3 has a natural automorphism σ of order 2 which sends agraded module (Mij)ij to (Mji)ij . and which exchanges the x and y action. LetC4 be the category of Z/2Z equivariant objects of C3, i.e., pairs (M,φ) where φis an isomorphism M −→ σ(M) satisfying σ(φ)φ = idM . Properties (C1)-(C4)follow easily from the fact that they hold for C3. In particular, the Serre functoris obtained from the one on C3 in the obvious way. As described, this constructionrequires char k 6= 2, but this can be circumvented [RVdB02].

We now present the most subtle and exotic example.

(E5) Let Q be a connected locally finite quiver such that the opposite quiver hasno infinite oriented paths. For a vertex x ∈ Q we have a corresponding projectiverepresentation Px and an injective representation Ix and by our hypotheses Ix isfinite dimensional. Let rep(Q) be the finitely presented representations of Q. Thecategory of all representations of Q is hereditary and hence, rep(Q) is an abeliancategory. It is easy to check that Ix ∈ rep(Q). Hence, the functor Px −→ Ix maybe derived to yield an endo-functor F : Db(rep(Q)) −→ Db(rep(Q)). This functorF behaves like a Serre functor in the sense that we have natural isomorphisms as inrelation [26], but unfortunately F need not be an autoequivalence. However, thereis a formal procedure to quasi-invert F so as to obtain a true Serre functor (see[RVdB02]). This yields a hereditary category rep(Q) which satisfies Serre duality.Under a technical additional hypothesis (see [RVdB02]) C5 = rep(Q) turns out tobe noetherian.

We would just like to recall a simple definition here.

Definition 3.21. (connectedness of an abelian category)An abelian category G is connected if, whenever G = G1⊕G2, then either G1 = G

or G2 = G.

The examples (E1)-(E5) are all connected and thanks to Reiten and M. vanden Bergh we know that they are the only hereditary ones satisfying (C1),(C2)and (C4).


Theorem 3.22. [RVdB02] Let C be a connected noetherian Ext-finite hereditarycategory satisfying Serre duality over k. Then C is equivalent to one of the categories(E1)-(E5).

It would be rather hasty to conclude that we have hit upon the right set ofhypotheses for a non-commutative analogue of a smooth proper variety. In fact,one indication that (C1)-(C4) may be inadequate is that, for example, they donot distinguish between algebraic and analytic (smooth compact) surfaces. Toget around this problem we resort to another property of smooth proper schemesthat was also discovered by Bondal and Kapranov and is not satisfied by analyticsurfaces.

Definition 3.23. (finite type cohomological functor)A cohomological functor H : Db(C) −→ mod(k) is of finite type if, for any

A ∈ Db(C),∑

n

dimkH(A[n]) <∞.

Then the appropriate condition is:

(C5) Let C be an Ext-finite abelian k-linear category of finite homological dimen-sion. Then C is saturated if every cohomological functor H : Db(C) −→ mod(k) offinite type is representable. [The condition requires that C be saturated].

It was shown in [BVdB03] that Coh(X) is saturated when X is a smooth projec-tive scheme and that saturation also hold for categories of the from mod(Λ) whereΛ is a finite dimensional algebra. Further, saturated categories satisfy Serre duality.So it is a stronger criterion.

Combined with theorem [3.22], this gives a much more compact classification.

Corollary 3.24. Assume that C is a saturated connected noetherian Ext-finitehereditary category. Then C has one of the following forms:

(1) mod(Λ) where Λ is an indecomposable finite dimensional hereditary algebra(this is a very special case of (E5)).

(2) Coh(O) where O is a sheaf of hereditary OX orders (see (E1)) over a smoothconnected projective curve X.

Remark 3.25. It can also be shown that the abelian categories occurring in thiscorollary are of the form qgr(R) for some graded ring R of GKdim 6 2. This, ina sense, will prove the classical commutative result that smooth proper curves areprojective.

We are now going to provide some criteria for a module category to be saturated.Interestingly, the seemingly strange χ conditions show up here as well. For theresult stated below, we declare that a connected graded ring R has finite rightcohomological dimension, provided the higher right derived functors of the functorLim−→

HomR(R/R>n,−) vanish i.e., RiLim−→

HomR(R/R>n,−) = 0 for i≫ 0 [refer to

the discussion on the dimension of Proj R].

Theorem 3.26. [BVdB03] Let R be a connected graded noetherian ring satisfyingthe following hypotheses:

1. R and its opposite ring R satisfy χ;


2. R and R have finite right cohomological dimension;3. qgr(R) has finite homological dimension.

Then qgr(R) is saturated.

As a justification for bringing in this saturation criterion we mention the followingresult.

Proposition 3.27. [BVdB03] Assume that X is an analytic K3 surface with nocurves. Then Coh(X) is not saturated.

With all the rambling above we hope to have augmented our understanding ofnon-commutative smooth proper “curves” at least. Now it is worth taking a closerlook at Serre duality as it only makes sense for categories which have finite homo-logical dimension, whereas in classical algebraic geometry much of the significanceof Serre duality lies in its applications to singular varieties, whose categories ofcoherent sheaves are infinite dimensional. So we investigate a second notion whichis applicable more generally.

Suppose that C is an Ext-finite noetherian k-linear abelian category with a dis-tinguished object O ∈ Ob(C). Then (C,O) is said to satisfy classical Serre dualityif there exists an object ω ∈ Db(C) together with a natural isomorphism

RHom(−, ω) ∼= RHom(O,−)∗

We can always pass on to Ind(C) to compute these derived functors. Since ω

represents a functor it is clear that it must be unique if it exists, and, moreover, itsexistence forces Hom(O,−) to have finite cohomological dimension. The followingresult gives us a class of rings for which ω exists.

Theorem 3.28. [YZ97b] Let (C,O) = (qgr(R), πR) for a noetherian connectedgraded ring R. Assume, further, the following:

1. R and its opposite ring R satisfy χ;2. R and R have finite right cohomological dimension.

Then (qgr(R), πR) satisfies classical Serre duality.

Remark 3.29. If we allow ω to lie inD(QGr(R)), then its existence does not requirethe χ assumption (see [Jør97]). Jørgensen, Yekutieli and Zhang among othershave explored the veracity of certain homological results from the commutativepurview in this new model under the assumptions of theorem [3.28] (see [Jør10],[Jør00], [JZ00], [SZ94b], [WZ00] and [YZ97a]). As a final remark it should bementioned that Gorenstein rings of finite left and right injective dimensions satisfythe assumptions of theorem [3.28] and hence, classical Serre duality (see [YZ97a]).

Generally one begins with definitions but we would like to bring the curtaindown on this section with a proposed definition of non-commutative P2 by Artinand Schelter, which uses bits and pieces of almost everything that we have discussedso far.

Definition 3.30. (non-commutative P2) ([SVdB01] definition 11.2.1.)A non-commutative P2 is a k-linear Grothendieck category of the form QGr(A),

where A is an AS-regular algebra of dimension 3, with Hilbert series (1 − t)−3.


Following the classification of such algebras described in chapter 1, one gets twodifferent P2’s, one of them being the known commutative one and the other, a trulynon-commutative P2.


4. Non-commutative Geometry via Deformation Theory

This line of thought was propounded by Laudal and it provides a different flavourfrom what we have seen so far. Still being very algebraic in its approach, it managesto develop an interesting “infinitesimal” theory, which is also in keeping with thetitle of the chapter. It is deemed fit that we take a quick look at the commutativedeformation theory after Schlessinger [Sch68].

Let k be an algebraically closed field. Let Artk denote the category of Artinian,local “commutative” k-algebras having residue field k i.e., diagrams of the form

k //

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

(A, m)

A/m = k

We shall denote Artk by C and by C we shall understand the pro-completionof C [it is dual to the construction of Ind(C) (see [22])]. C has a more intelligibledescription: its objects are complete local k-algebras R, such that R/mn ∈ C for alln > 1, where m is the unique maximal ideal of R. A functor F : C −→ Set naturallyextends to a functor F : C −→ Set and by some Yoneda kind of argument we have

F (R) ≃ Hom(hR, F )

So ζ ∈ F (R) induces a morphism hR −→ F .

Definition 4.1. (Smoothness of morphisms of functors)A morphism F −→ G of functors is smooth if for any surjection B −→ A in C

the morphism

F (B) −→ F (A) ×G(A) F (B)

is surjective as well.

The tangent space of F , denoted by tF := F (k[ǫ]).

Definition 4.2. (Pro-representing hull)

Let F : C −→ Set be a functor. A couple (R, ζ), where R ∈ C and ζ ∈ F (R),is a pro-representing hull of F if the induced map hR −→ F is smooth and if inaddition thR −→ tF is a bijection.

Analogously one would say that (R, ζ) pro-represents F if the morphism hR −→ Finduced by ζ is an isomorphism.With all these definitions in our fingertips, we can now state the following theoremdue to Schlessinger [Sch68].

Theorem 4.3. Let F : C −→ Set be a covariant functor such that F (k) is singleton.Let A′ −→ A and A′′ −→ A be morphisms in C and consider the map,

F (A′ ×A A′′)

ψ−→ F (A′)×F (A) F (A′′)

Then F has a pro-representing hull if and only if F has the properties (H1),(H2) and (H3) below:


(H1) ψ is a surjection whenever A′′ −→ A is a small extension [i.e., ker(A′′ −→ A)is a principal ideal ⊂ Ann(mA′′)](H2) ψ is a bijection when A = k, A′′ = k[ǫ].(H3) dimk tF <∞.

Further, F is pro-representable by a couple if and only if F satisfies(H4) ψ is an isomorphism whenever A′ = A′′ and A′ −→ A is a small extension.

Let X ∈ Schk and A ∈ C = Artk. Then by deformation of X to A we mean adiagram of the form

Xi

−−−−−−−−−−−→closed immersion

Yy

yflat

Spec k −−−−→ SpecA

such that X∼−→ Y ×SpecA Spec k. To any morphism A −→ B and Y , a defor-

mation of X to A, we can associate Y ×SpecA SpecB, a deformation to B. We canpackage all these information into one single definition.

Definition 4.4. (Deformation functor DefX|k)

DefX|k :Artk −→ Set

A 7−→ set of isomorphism classes of deformations of X to A

Schlessinger in [Sch68] showed that DefX|k satisfies (H1) and (H2). For (H3)he needed some assumptions on X (e.g., X is proper over k). For (H4) also he hasgot a requirement which we are going skip over here.

One can easily figure out the right definition of the deformation functor of otheralgebraic geometric objects e.g., sheaves of modules. With that we come to the endof our cruise through commutative “deformation theory”.

We shall look at deformations which will eventually enable us to enter the worldof non-commutative algebraic geometry. With some foresight, we shall study thesimultaneous formal deformations of a finite family [the infinite theory can be de-veloped suitably as a limiting case of the finite ones] of right A-modules, where A isan associative k-algebra. The right place i.e., category to develop non-commutativedeformation theory turns out to be ar, which is a suitable sub-category of Ar, thecategory of r-pointed k-algebras. The objects of Ar are the diagrams of k-algebrasof the following form:

kri

//

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

R

p

kr

A few words about terminology: by a diagram we shall understand an orientedgraph (could be infinite) with possibly multiple edges between two vertices. Thisis essentially a local terminology. There is no more assumption on a diagram andso such an object may not be a category (e.g., if it does not have any self loop ata vertex, which is by the way not forbidden, then the identity morphism does notbelong to the diagram for that particular vertex). Since only the map p is needed


in most cases, henceforth we shall also denote the objects of Ar by p : R −→ kr orsimply by R.

The radical of R is Rad(R) := ker(p). Put TR = (Rad(R)/Rad(R)2)∗ and callit the tangent space of R. The category ar is the full sub-category of Ar consistingof objects in which R is Artinian and complete in the Rad(R)-adic topology.

Fix a finite diagram V = V1, . . . , Vr of right A-modules. Such a family is calleda swarm if, for all i, j

dimkExt1A(Vi, Vj) <∞

Henceforth we are going to use the terms, “diagrams” and “swarms”, almostinterchangeably. For any R ∈ ar we define a lifting of V to R as an R−A bimoduledenoted by VR, together with isomorphisms ηi : ki⊗RVR −→ Vi of right A-modulesfor 1 6 i 6 r, such that the left and right k-vector space structures coincide andabstractly VR ≃ R ⊗k V as left R-modules. The requirement that VR ≃ R ⊗k Vas left R-modules generalizes the flatness condition of commutative deformationtheory.

Let V ′R,V′′R be two liftings of V to R. We say that these two liftings are isomorphic

if there exists an isomorphism τ : V ′R −→ V ′′R of R − A-bimodules, such thatη′′i (Id ⊗ τ) = η′i for all 1 6 i 6 r. We refer to these equivalence classes asdeformations of V to R. Although ηi’s form an integral part of the definition ofliftings and deformations, for convenience we shall often suppress them.

Definition 4.5. (Non-commutative deformation functor)It is a covariant functor defined as follows:

DefV :ar −→ Set

R 7−→ set of deformations of V to R

Given any morphism R −→ S in ar one can easily check that VS = S ⊗R VR isa deformation of V to S, which is independent of the lifting chosen.

Now it is a theorem due to Laudal, which illustrates the pro-representability ofthe non-commutative deformation functor.

Theorem 4.6. The functor DefV has a pro-representable hull i.e., an object H(V) =H ∈ ar, together with a versal family,

V = H⊗kV

such that the corresponding morphism of functors in ar,

ρ : Homar(H,−) −→ DefV

is smooth (see [4.1]) and an isomorphism at the tangent level [i.e., ρ is an iso-morphism for all R ∈ ar for which Rad(R)2 = 0].

The fairly involved proof of this theorem can be found in [Lau02]. Ile has a prooftoo, which is actually closer in spirit to that of Schlessinger in the commutativecase (see [Ile90]). One more apposite remark has to be made before we move on.


Remark 4.7. By virtue of the definition of a deformation, V is itself an H − A-bimodule. The right A-module structure of V defines a homomorphism of k-algebras,

η : A −→ EndH(V ) =: O(V)

and the k-algebraO(V) extends the right module action ofA on V = V1, . . . , Vr.

A generalized Burnside theorem: Let A be a finite dimensional k-algebra,k being an algebraically closed field. Suppose that the family V = V1, . . . , Vrcontains all simple A-modules, then

A ≃ O(V)

Further, A is Morita equivalent to H(V), which is evident from the relationbetween O(V) and H(V).

We have barely been able to scratch the surface of non-commutative deformationtheory, and it is time to leave it at that and plunge into non-commutative algebraicgeometry. Consider X := SpecA, where A is a commutative finite type k-algebra.Then the closed points of X are given by the maximal ideals and to any such closedpoint x ∈ X we can associate the A-module k(x) which is the residue field at x.Suppose for V we take the singleton set k(x) for a fixed x ∈ X . It is almosttautological to say that X is the moduli space of its closed points (not all points)and one has to ruminate a little to be convinced of the fact that H(V), the hull of

the deformation functorDefV , will be given by Amx , which is the mx-adic completionof the local k-algera, OX,x = Amx . A section f of the structure sheaf can be read off

from its germs fx ∈ Amx at the different points x ∈ X . This is what we are goingto cash in on and the new notion of structure sheaf will be obtained from germanedeformations.

For non-commutative rings we have a good cohomology theory, namely HochschildCohomology (and also Connes’ Cyclic cohomology). In the evolutionary process ofmathematical objects, rings seem to be the ancestors of pre-additive categories andnow we shall see the evolution of the cohomology theory which will be applicableto small abelian categories. By the way, evolution is a dynamic process.

For any small abelian categore C we shall define a new category MorC.1. The objects are just morphisms in C2. If φ, φ′ are two arrows in C then HomMorC(φ, φ

′) is the set of commutativediagrams

∗ψ

←−−−− ∗

φ

yyφ′

∗ −−−−→ψ′

∗

We shall simply write (ψ, ψ′) for such an arrow in MorC.Now consider any covariant functor G : MorC −→ Ab. We shall define a chain

complex, D∗(C, G) as follows:


D0(C, G) =∏

c∈Ob(C)

G(Idc)

Dp(C, G) =∏

c0ψ1→c1...

ψp→cp

G(ψ1 · · · ψp)

where the indices are strings of morphisms ψi : ci−1 −→ ci of length p. Thedifferential dp : Dp(C, G) −→ Dp+1(C, G) is defined in the obvious manner i.e., ifζ ∈ Dp(C, G) then the ψ1 · · · ψp+1th entry of dp(ζ), denoted by dp(ζ)(ψ1,...,ψp+1)

is:

dp(ζ)(ψ1,...,ψp+1) =G(ψ2 · · · ψp+1 → ψ1 · · · ψp+1)(ζ(ψ2,...,ψp+1))

+

p∑

i=1

(−1)iζ(ψ1,...,ψiψi+1,...,ψp+1)

+ (−1)p+1G(ψ1 · · · ψp → ψ1 · · · ψp+1)(ζ(ψ1, . . . , ψp))

where (ψ2 · · · ψp+1 → ψ1 · · · ψp+1) and (ψ1 · · · ψp → ψ1 · · · ψp+1) arearrows in MorC.

Now one has to check that dp+1 dp = 0 and then we may summarize our effortsby saying that

D∗ : Funct(MorC, Ab) −→ Cat. of complexes of ab. gps.

G 7−→ (Dp(C, G), dp)p>0

is a functor, which has been shown to be exact in [Lau79]. Note that, the functorD∗ is exact, but given a functor G : MorC −→ Ab, the complex (Dp(C, G), dp) canhave non-trivial cohomologies.

We define the cohomology of the category C with respect to a functor G :MorC −→ Ab as

H∗(C, G) := H∗(Dp(C, G), dp)

Given an associative k-algebra A, let C be any sub-diagram of Mod(A), thecategory of right A-modules. Let π : C −→Mod(k) denote the forgetful “functor”.Then we can manufacture a functor Homπ : MorC −→Mod(k).

Homπ : MorC −→Mod(k)

c1ψ→ c2 7−→ Homk(π(c1), π(c2))

This allows us to define a transient object, which is Oo(C, Homπ) := H0(C, Homπ).I am afraid, but we are going to come across a plethora of O’s now. We shallsimplify notation by writing just Oo(C) instead of Oo(C, Homπ). It is clear fromthe definition of the cohomology that Oo(C) ⊂

∏c∈C

Endk(π(c)) and it inherits its

algebra structure from∏c∈C

Endk(π(c)). Moreover, it also has a canonical projection

onto each Endk(π(c)) making c into an Oo(C)-module. There is also a canonical


homomorphism of k-algebras η0 : A −→ Oo(C) as every a ∈ A is seen to give riseto a 0-cocycle.

We had started with C, which was a sub-diagram of Mod(A), and we haverealized that each c ∈ C is also an Oo(C)-module. So we may play the same gameconsidering C as a sub-diagram of Mod(Oo(C)) and taking a new forgetful functor(restricting to C)

π0 : Mod(Oo(C)) −→Mod(k)

The striking thing is that Oo(C, π0) ∼= Oo(C, π) and that we get nothing new.

Remark 4.8. For a commutative k-algebra A of finite type let Prim(A) denote thesub-category of Mod(A) comprising the indecomposable modules of the form A/qfor some primary ideal q. Then, if A = k[ǫ], it turns out that Oo(Prim(A), π) ≃2×2 upper triangular matrices with entries in k. Here, the algebraA has nilpotentelements. However, if A were reduced we would have had an isomorphism via η0.

There is a Jacobson topology on the set Prim(A) (A reduced) defined as follows:Let a ∈ A and consider the subset D(a) of Prim(A) defined by the objects M

for which a /∈ Ann(M). Then clearly D(a) ∩ D(b) = D(ab) and hence, they forma basis of the topology generated by them. What is more interesting is that, D(a)is simply the localization of Prim(A) at the multiplicative subset generated by thepowers of a and for each a there is a canonical isomorphism

Oo(D(a), π) ≃ A(a) = OSpec A(D(a))

In fact, as schemes SpecA and Prim(A) are isomorphic. So Jacobson topologyis a reincarnation of Zariski topology over Prim(A). One could have thought ofdefining Prim(A)’s as non-commutative schemes but, actually there is still roomfor improvement.

Recall the O(V) construction (see [4.7]) for a swarm V of A-modules. It cameequipped with a canonical k-algebra homomorphism η : A −→ O(V). Also fromthe very definition of the terms, we obtain a canonical k-algebra homomorphismρ0 : O(V) −→ Oo(V). We summarize the whole state of affairs in the followingcommutative diagram for a finite diagram V of A-modules.

Aη

//

η0!!D

D

D

D

D

D

D

D

D

O(V)

ρ0

Oo(V)

The arrows extend the module actions on V .

Now we introduce a notation which should have been done much earlier. Forany diagram of A-modules V := V1, . . . , Vr let |V| denote the underlying setof modules i.e., the diagram stripped off its morphisms. So far we have not atall made use of the fact that there could be morphisms between the deformedmodules of the swarm (or the diagram) and so the O(V) that we have constructedis actually O(|V|). There is a way to incorporate the incidence datum of the swarmof modules into the definition of the algebra O(V), which we are unfortunatelynot able to discuss. However, we just present the definition with a host of strange


symbols with an appeal to the interested readers to look them up in [Lau00]. Thenomenclature has a direct relationship with physics.

Definition 4.9. (Algebra of pre-observables)The k-algebra of pre-observables O(V , π) of the finite swarm V, is the sub-algebra

of (H(V) ⊗ Endπ(V)) [generated by the morphisms in πV] commuting, for anyS ∈ ar, via the morphism

κS : (H(V)⊗ Endπ(V)) −→ (S ⊗ Endk(V))(27)

induced by any surjective k-algebra homomorphism

H(V) −→ S

with the corresponding representation of the path algebra of V (thought of as aquiver) in (S ⊗ Endk(V)).

Now we need to take a look at the problem of functoriality. This O-constructionis not functorial with respect to the inclusions of diagrams of modules and theircorresponding path algebras. An arrow of the form k[C0] → k[C] induces an arrowlike the one below:

o(C0 ⊂ C) : (H(C)⊗ End(C)) −→ (H(C0)⊗ End(C0))

We now desire that there be a natural morphism

O(C ⊂ C0) : O(C, π) −→ O(C0, π)

which is not quite obvious from the construction of O. And so we get aroundthis problem by defining the algebra of observables, which morally should be thesmallest k-algebra extending the action of A on C and having all the nice functorialproperties. We just claim that the following definition works, which bluntly triesto fix the problem.

Definition 4.10. (algebra of observables)The k-algebra of observables of the finite swarm C is the sub-algebra

O(C, π) :=⋂

C0⊂C

o(C0 ⊂ C)−1(O(C0, π)) ⊂ O(C, π)

where C0 ⊂ C runs through all sub-diagrams of C. Here o(C0 ⊂ C)−1(O(C0, π))denotes the set-theoretic inverse image.

Now following [Lau00] we claim that this new O-construction of the algebraof observables is a contravariant functor on the partially ordered category of sub-diagrams of a given diagram C.

We need to extend the construnction to infinte diagrams. We call a swarm Cpermissible if there exists a k-algebra homomorphism,

η(|C|, π) : A −→ O(|C|, π),

compatible with the morphisms η(|C0|, π) and o(C0 ⊂ C), where C0 runs throughall sub-diagrams of C.


Now for a permissible (possibly infinite) swarm C we define

O(C, π) := Lim←C0⊂C

O(C0, π)

where C0 runs through all finite sub-diagrams of C.Finally we arrive at the notion of the structure sheaf Oπ. For every finite sub-

diagram C0 of a swarm C, consider the natural morphism

κ(C0) : O(C, π) −→ (H(C0)⊗ End(C)) refer to [4.7] and [27]

and consider the two-sided ideal n ⊂ O(C, π), defined by

n =⋂

C0⊂C

kerκ(C0)

where C0 once again runs through all finite sub-diagrams of C. We now set

Oπ(C) = O(C, π)/n

Going through all these rigmarole should have some tangible benefit and that isgiven by the following definition of an affine non-commutative scheme.

Definition 4.11. (Affine non-commutative scheme)A swarm of A-modules, C, is called an affine scheme for A, if

η(C) : A −→ Oπ(C)

is an isomorphism and we shall refer to A as the affine ring of this scheme.Note that we are talking about “an” affine scheme for A and not “the” affine schemeas there could be several different scheme structures for A.

Remark 4.12. For a commutative k-algebra A (not necessarily finite type) let

Simp(A) denote the set of simple A-modules. It is seen that O(Simp(A)) ≃∏m

Am,

where m runs through all maximal ideals of A. However, η(Simp(A)) fails to be anisomorphism between A −→ O(Simp(A), π) in general. This can be rectified byjust including A in Simp(A), which we denote by Simp∗(A).

Infinitesimal Structures

Let C be a swarm ofA-modules and consider the point x = Vi ∈ C. Set V = ⊕V ∈C

V .

Definition 4.13. (Tangent space)The tangent space at x, denoted by Tx is defined to be

Tx := ξ ∈ Ext1A(Vi, Vi) | ∀p ∃ξp such that

∀φ := φi,p : Vi −→ Vp φ∗(ξ) = φ∗(ξp) and

∀φ := φp,i : Vp −→ Vi φ∗(ξp) = φ∗(ξ)


It can be verified that the composition of the natural maps Derk(A,A) −→Derk(A,Endk(Vi)) and the surjection Derk(A,Endk(Vi)) −→ Ext1A(Vi, Vi) actu-ally gives a map

θx : Derk(A,A) −→ Tx ⊂ Ext1A(Vi, Vi)

Definition 4.14. (Smoothness)We call a point x = Vi ∈ C smooth if the map θx is surjective. If this is true for

all points of C, we say that C is smooth.

The author also has a short treatment of invariant theory and moduli at theend of [Lau00]. All this is not futile but, due to lack of time, we need to endabruptly. The most interesting scheme structure for an algebra A is probably givenby Simp∗(A), which has been discussed at length in [Lau00]. These seeminglypreternatural definitions have some interesting and revealing applications whichcan be found in [LS04].


5. Synopsis of some other points of view

This section is really meant to be a glossary rather than an in-depth analysis.In spite of that, given the staggering number of articles in this area, we shall onlybe able to see the tip of the iceberg. It is worth mentioning that curious readersmay take a look at the web-page of Paul J. Smith

http://www.math.washington.edu∼smith/Research/research.html

where one can find links to a host of other interesting web-sites catering to researchin non-commutative algebra and algebraic geometry.

1. The first attempt that I would like to talk about is Noncommutative spacesand flat descent by Kontsevich and Rosenberg and it uses absolutely “state-of-the-art machinery” (refer to [KR04]). In a previous article entitled “Noncommutativesmooth spaces” (see [KR00]) the same authors had advocated two principles innon-commutative geometry. One principle asserts that non-commutative notionsshould come from the representations of their commutative counterparts and theother says that it should be possible to read off a non-commutative object from its“covering datum”. The article under consideration improves upon the second line ofthought, but it is much more abstract in nature. This approach adheres to sheavesof sets on the category of affine schemes endowed with flat topology and havingdescent property. The minimum desideratum for non-commutative geometry, orfor that matter any geometric theory, is a category of local or “affine” objects,and a functor which to every such object assigns a “space”, like Spec. This articlerelies on the philosophy that a “space” can be identified with categories, which arethought of as categories of coherent or quasi-coherent sheaves on it. So the authorscarry along with each “space” X a category CX , which is regarded as the categoryof quasi-coherent sheaves on X . So it is a mathematical object like a fibred category

C

π

y

Bwhere the base category B serves as the category of “spaces”. For any X ∈ B

we denote by CX the fibre category over X , which is thought of as the category

of quasi-coherent sheaves on X . For any morphism Xf−→ Y we denote by f∗ the

corresponding 2-morphism CY −→ CX and call it the inverse image functor of f .Now we set C = Catop and B = |Cat|0 where Cat stands for the 2-category of

categories and we set about defining |Cat|0.

Ob(|Cat|0) = objects of Cat i.e., categories.Hom|Cat|0(X,Y ) = isomorphism classes of functors from CatopY −→ CatopX .

The authors write f = [F ] to indicate that f is a morphism such that the functorF belongs to the isomorphism class of its inverse image functors. Define a functorfrom Catop −→ |Cat|0, which is identity on the objects and which sends each

morphism CatopYF−→ CatopX to the morphism X

[F ]−→ Y , to make Catop into a fibred

category over |Cat|0. The authors then orchestrate a definition of a “cover”. Usingthese covers they are able to define affine objects and gluing of such objects to formlocally affine objects, which can be recovered uniquely from the covering data via

http://www.math.washington.edu~smith/Research/research.html


“flat descent”. To satisfy one’s curiosities, interested readers are encouraged to gothrough this article.

2. Now we are going to talk about Noncommutative Geometry Based on Commuta-tor Expansions by M. Kapranov (refer to [Kap98]). The author himself describes itas an attempt to develop non-commutative algebraic geometry “in the perturbativeregime” around ordinary commutative geometry. Only algebras over the complexnumbers are considered. So given any associative C-algebra R, a decreasing filtra-tion of R, denoted by F dRd>0 is defined as follows:

F dR =∑

m

∑

(i1+···+im−m=d)

R.RLiei1 .R . . . R.RLieim .R

where RLie is the Lie algebra concocted from the associative algebra R. Thena suitable notion of NC-completion (R −→ RNC−complete = Lim

←R/F dR) of such

algebras is defined and for an NC-complete algebra R he is able to define thelocalization for any multiplicatively closed T ⊂ Rab (Rab := R/[R,R]) as:

R‖T−1‖ = Lim←

(R/F d+1R)[T−1]

For everyNC-nilpotent algebra a space with a sheaf of functionsX = Spec (R) =(SpecRab,OX) can be defined as follows:

OX(Dg) = R[g−1]

where Dg = ℘ ∈ X |g /∈ ℘. And the stalks are defined as

OX,℘ := R℘ = Lim→

℘∈Dg

R[g−1]

Set Xab = SpecRab. Then, for an NC-complete algebra R, the formal spectrumSpf(R) is given by the ringed space (Xab,OX), where OX is the sheaf of topologicalrings obtained as the inverse limit of the structure sheaves of Spec (R/F d+1R).

The ringed space Spf(R) is called an affine NC-scheme andNC is the category ofNC-complete algebras and, as expected, there is indeed an equivalence of categoriesbetween the category of NC-schemes and NC given by X −→ Γ(Xab,OX). Thestory does not end here. Given an ordinary manifold M , he considers objects(thickenings) X with Xab = M and towards the end illustrates that several familiaralgebraic varieties, including the classical flag varieties and all the smooth modulispaces of vector bundles, possess natural NC-thickenings. In addition, notions likeNC-smoothness are discussed and, conforming to our expectations, for an NC-smooth algebra R, dimR = n, if x is a C-valued point of SpecRab, it is shown thatRx ≃ C〈〈x1, . . . , xn〉〉.

3. An book published in 1981 entitled “Noncommutative Algebraic Geometry” byOystaeyen and Verschoren (refer to [OV81]) promulgates a theory which works wellfor affine PI-algebras over an algebraically closed field k. It heavily relies on theconcomitant developments in non-commutative ring theory, carried out by Artin,Schelter, Cohn, Procesi and the authors themselves, to name only a few. One of


the watermarks of the book is the non-commutative version of Riemann-Roch forcurves.

4. We may “derive” the philosophy of identifying a space with its category ofquasi-coherent sheaves. Ever since the article [Beı84] by Beılinson describing thederived category of coherent sheaves on Pn there has been a spate of activities onsuch things. We would like to point out particularly the article entitled “Derivedcategories of coherent sheaves” by Bondal and Orlov which ends with some intrigu-ing connections with non-commutative geometry (refer to [BO02]). They try to dosome kind of birational geometry in this derived set-up and Orlov has taken thisnotion one step further with his article [Orl03a]. Grothendieck’s notions of a “site”and a “topos” hold the promise of being generalized to the non-commutative frame-work and it seems that something similar has been achieved by Orlov in [Orl03b].

5. A highly readable article by Pierre Cartier [Car01] entitled “A MAD DAY’SWORK: FROM GROTHENDIECK TO CONNES AND KONTSEVICH. THEEVOLUTION OF CONCEPTS OF SPACE AND SYMMETRY” provides a sub-lime description of the evolution of our notions of space and symmetry. It also dis-cusses some connections between non-commutative geometry via topos theory afterGrothendieck and that of Connes using operator algebras (He refers to [Tap91] in theprocess). Interestingly, it begins with a short biography of Alexander Grothendieckand end with “a dream” of Cartier. It cannot be totally classified as an article innon-commutative algebraic geometry but the whole treatise is not only instructivebut edifying.

Finally we just mention some other sources of knowledge in this area. There is aset of notes by Le Bruyn [Bruyn] which can be downloaded freely from his web-site.Smith and Zhang had disseminated a nice approach in [SZ98] of studying curves innon-commutative schemes. Lately Polishchuk has come up with some interestingideas for which two good sources could be [Pol04b] and [Pol04a]. M. Van denBergh has an article entitled “Blowing up of non-commutative smooth surfaces”[VdB01] and he has followed it up with a recent article in the arXiv server, dealingwith blowing down [VdB98]. Incidentally Keeler, Rogalski and Stafford also havetheir own version of blowing up in [RS03]. Last and perhaps the closest to beingdefinitive article providing an overview of the goings-on in this area is [SVdB01]by Stafford and Van den Bergh, which was also the backbone of this write-up.Recently Matilde Marcolli has written a book entitled “Arithmetic NoncommutativeGeometry” [Mar05], which explores some interesting and intriguing connectionsbetween non-commutative geometry and arithmetic.

There are also some examples of non-commutative spaces like D-schemes ofBeılinson-Bernstein and non-commutative schemes of P. Cohn which readers areinvited to unravel for themselves.

Noncommutative geometry is now a rich and widely pursued subject. If naturedesires to divest herself of commutativity, as examples coming from physics man-ifest, then it may not be as “natural” as we think. One question that was mulledby several people is whether noncommutative geometry is just what we obtain oncewe purge the commutativity hypothesis. Also, is non-commutative geometry justa shadow of its more resplendent predecessor? There is room for arbitration but


clearly, it has had some impact on our perception of a “space” and it will continueto engage us in its myriad problems.


6. ACKNOWLEDGEMENTS

So far a meticulous reader must have noticed that the first person singular num-ber was very carefully avoided. However, here I would just be myself and do thehonours.

First of all I would like to thank my advisor, Prof. Matilde Marcolli, for in-troducing me to this subject and lending me a patient ear during the discussionsessions that we had. She was also instrumental in helping me attend a few schoolsand conferences, which have substantially augmented my knowledge, and editedmy write-up on quite a few occasions.

I would also like to thank some of my friends, particularly Jorge Andres PlazasVargas and Eugene Ha in Max-Planck-Institut fur Mathematik, Bonn who tooktheir time out of their busy schedules to take part in our regular discussion sessionsand often providing insightful remarks.

A would specially like to thank Prof. Michel Van den Bergh for going throughthe first two chapters and providing valuable comments, suggestions and sometimeseven graphic explanations to my queries. Even while writing the later chapters wecommunicated via e-mail and he was always forthcoming with his answers, howeversilly my questions might have been.

It would be thoroughly unfair if I fail to acknowledge the people who have proofread this article at various stages of its development or helped me in LaTeX-ing,namely Gunther Vogel and Ozgur Ceyhan. Gunther had also provided me withsome useful references.

Finally, I take this opportunity to extend my earnest gratitude to everyone atMax-Planck-Institut fur Mathematik, Bonn, for providing a very conducive envi-ronment for studies.

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Max-Planck-Institut fur Mathematik, Bonn, Germany.

E-mail address: [email protected]

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